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Icarus 268 (2016) 156–171
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Icarus
journal homepage: www.journals.elsevier.com/icarus
Polarimetry of moonlight: A new method for determining the refractive
index of the lunar regolith
Andrew Fearnside a,⇑, Philip Masding b, Chris Hooker c
a
No. 2 The Cobbles, Boots Green Lane, Allostock, Knutsford, Cheshire WA16 9NG, UK
Imber House, Vale Road, Bowdon, Cheshire WA143AQ, UK
c
6 High Street, Didcot, Oxfordshire OX11 8EQ, UK
b
a r t i c l e
i n f o
Article history:
Received 31 March 2015
Revised 10 October 2015
Accepted 30 November 2015
Available online 29 December 2015
Keyword:
Polarimetry
Moon, surface
Regoliths
Mineralogy
a b s t r a c t
We present a new method for remotely measuring the refractive index of the lunar regolith, using
polarised moonlight. Umov’s Law correlates the polarisation (Pmax) of scattered moonlight and the albedo
(A) of the scattering lunar regolith. We discuss how deviations from this correlation have previously been
linked to the so-called ‘Polarimetric Anomaly Parameter’, (Pmax)aA, which was proposed by Shkuratov and
others as being related to variations in regolith grain size. We propose a reinterpretation of that parameter. We develop models of light scattering by regolith grains which predict that variation in the refractive index of regolith grains causes deviations from Umov’s Law. Variations in other grain parameters
such as grain size and degree of space weathering do not produce this deviation. The models are supported by polarimetric measurements on powdered terrestrial materials of differing refractive index.
We derive a simple formula to express the relationship between refractive index and the deviation from
Umov’s Law and apply it to telescopic measurements of regions of the lunar surface. We show that the
Aristarchus Plateau and the Marius Hills regions both comprise materials of unusually low refractive
index. These results are consistent with recent estimates of the mineralogy of those areas. Picard and
Peirce craters, in Mare Crisium, are shown to contain material of low refractive index similar to highland
regions, as has been suggested by earlier studies of these craters.
Ó 2015 Elsevier Inc. All rights reserved.
1. Background
Moonlight is linearly polarised at almost all lunar phase angles.
The degree of linear polarisation is defined as P = (I\ I||)/(I\ + I||)
where I\ and I|| are the intensities of the components of moonlight
with electric field vectors resolved perpendicular (\) or parallel (||)
to the scattering plane. This quantity varies with lunar phase angle
and reaches a peak (Pmax) at lunar phases typically lying between
about 95° and about 110°. This peak value correlates strongly with
the albedo (A) of the lunar surface region in question. The correlation, known as Umov’s Law, is linear when displayed on logarithmic axes and may be described by an equation of the form:
log10(A) + a1log10(Pmax) = a2. Dollfus and Bowell (1971) applied this
regression relation to observations of over a hundred small regions
across the lunar surface. In the visual spectral range of light, they
found that a1 = 0.72 and a2 = 1.81. Deviations from Umov’s Law
caught the interest of Shkuratov, Dollfus and others (Shkuratov,
1981; Dollfus, 1998; Shkuratov and Opanasenko, 1992; Geake
⇑ Corresponding author.
E-mail address: [email protected] (A. Fearnside).
http://dx.doi.org/10.1016/j.icarus.2015.11.038
0019-1035/Ó 2015 Elsevier Inc. All rights reserved.
and Dollfus, 1986; Novikov et al., 1982) who studied this law in
relation to granular terrestrial materials and lunar regolith samples
alike. When applied to moonlight, it was found that departures
from Umov’s Law occurred as a ‘polarisation excess’ when associated with the ejecta of some bright young craters, and as a ‘polarisation deficit’ notable on the Aristarchus Plateau (so-called
‘Wood’s Spot’) and the Marius Hills, for example. It was suggested
that a ‘polarisation excess’ was caused by anomalously large median grain sizes within the granular material being observed. The
‘Polarimetric Anomaly Parameter’: ðPmax Þa1 A was derived as a
means of quantifying departures from Umov’s Law and it was suggested that variations in this parameter quantified variations in the
median grain size of the lunar regolith. Shkuratov proposed that
‘polarisation deficit’ might result from grains of anomalously low
albedo causing a breakdown of Umov’s Law, though later work
suggested that lower median grain sizes might be the cause. The
question of how to interpret ‘polarisation excess’ and ‘polarisation
deficit’ has received very little attention since. Indeed, more recent
work (Shkuratov et al., 2007; Hines et al., 2008; Jung et al., 2014)
on lunar polarimetry has assumed the correctness of this interpretation of the ‘Polarimetric Anomaly Parameter’.
A. Fearnside et al. / Icarus 268 (2016) 156–171
In this paper we address this question. We propose a new interpretation of ‘polarisation deficit’ and ‘polarisation excess’. This new
interpretation is that departures from Umov’s Law are the result of
variations in the refractive index of lunar regolith grains. This
allows a new method for determining the refractive index of the
lunar regolith remotely.
2. The plan of this paper
We begin in Section 3 by presenting a simple 2-dimensional
(2D) model of light scattering from a lunar regolith grain. We show
how this model may be used to interpret the structure of a correlation obeying Umov’s Law, including a consistent interpretation of
both a ‘polarisation excess’ and a ‘polarisation deficit’. We then test
this simplistic 2D model by considering a 3-dimensional (3D)
multi-grain regolith model and show that it fully supports our
interpretation of ‘polarisation excess’ and ‘polarisation deficit’.
Next, in Section 4, we describe polarimetric experiments conducted on terrestrial grain samples and lunar regolith simulants.
We show that these experiments confirm our new interpretation
of ‘polarisation excess’ and ‘polarisation deficit’. Finally, in Section 5, we apply our interpretation to the polarimetry of moonlight.
We show how polarisation measurements can be applied to determine the refractive index of the lunar regolith. The Aristarchus Plateau and the Marius Hills are considered in detail, in order to allow
a comparison with the work of Shkuratov (1981) who considered
these areas in particular. Mare Crisium is also discussed in terms
of its stratigraphy as revealed by observational results presented
here.
3. Theory
Our simple 2D model represents an idealised regolith grain in
isolation. This model assumes that inter-grain scattering of light
has a negligible influence upon the polarisation properties of
moonlight. Our 3D model takes account of inter-grain scattering
and we will show that these two models provide a mutually consistent interpretation of Umov’s Law.
tive index of the grain core was based on values for pyroxene and
olivine stated by Hiroi and Pieters (1994), which include the necessary wavelength dependency. For npFe, the optical properties of
iron were taken from Johnson and Christy (1974). By applying
Maxwell–Garnett effective medium theory, we define:
3/ðK Fe K 2 Þ=ðK Fe þ 2K 2 Þ
K1 ¼ 1 þ
1 /ðK Fe K 2 Þ=ðK Fe þ 2K 2 Þ
3.1.1. Grain structure
A lunar regolith grain is represented as a core of uniform material of diameter D as shown schematically in Fig. 1. This core carries
a uniform coating of space-weathered core material defining a
shell of thickness d. The shell consists of a proportion (/) of
nano-phase metallic iron beads (npFe) embedded in the same
material as the core. The effective refractive index of the spaceweathered layer is represented by m1 = n1 + ik1 and is calculated
using the refractive index (m2 = n2 + ik2) of the core material and
the refractive index (mFe = nFe + ikFe) of the npFe beads. The refrac-
ð1Þ
Here, KFe = (nFe + ikFe)2, K2 = (n2 + ik2)2 and K1 = (n1 + ik1)2 from
which m1 is obtained. The absorption coefficients of the grain core
and space-weathered layer, for light of wavelength k, are defined
as a2 ¼ 4pk2 =k and a1 ¼ 4pk1 =k respectively. The absorption coefficient (a2) for the grain core was derived from graphs presented by
Nimura et al. (2008) concerning Apollo 16 soil samples. This type of
model has also been used by Nimura to represent the upper layer of
the lunar regolith (Nimura et al., 2008).
We calculate the Fresnel amplitudes ^rjj;?
ðhÞ for reflected light
ab
polarised perpendicular (\) or parallel (||) to the scattering plane
at each internal and external grain boundary, where h is the angle
of incidence of a ray traversing between medium ‘‘a” and medium
‘‘b”. Transmission and reflection coefficients were then calculated,
and applied in a manner suggested by Hapke (2005) to derive an
expression for the proportion of light scattered by the grain at a
given phase angle (g), and the degree of linear polarisation (P(g))
of that scattered light. To do this, we calculated the single scattering albedo (w) of the grain in terms of the total scattering (sab) and
transmission (tab) activities using angle-averaged values
R
ðr jj;?
¼ 2 j^rjj;?
ðhÞj2 cos h sin hdhÞ of the Fresnel amplitudes. These
ab
ab
are identified in Table 1 in terms of boundaries between different
media at or within the grain. The reader is referred to schematic
Figs. 2 and 3 presented by Nimura et al. (2008) which succinctly
show the light scattering processes being modelled here.
The total single scattering albedo is given by w = (w\ + w||)/2.
We calculated the degree of linear polarisation according to the
definition provided by Hapke (Chapter 14, Eq. (14.3) (Hapke,
2005)) in terms of bidirectional reflectance as:
PðgÞ ¼
3.1. 2D model
157
½X ? ðg=2Þ X jj ðg=2Þ
½X ? ðg=2Þ þ X jj ðg=2Þ þ 2½wH2 Y
ð2Þ
where g/2 is the angle of incidence and reflection. Here, the term Y
is the angle-averaged value of (X\(h) + X||(h)). The terms X\(h) and
X||(h) are the back-scattering activity expressed using anglex
x
dependent (i.e. not angle-averaged) reflectances ðRab
ðhÞ ¼ j^rab
ðhÞj2 ).
These are defined in Table 1 in terms of reflectances at boundaries
between different media at and within the grain, and they account
for back-scattered light from the space-weathered layer both externally at the grain surface and also internally, having performed one
round-trip through the grain core and back. The angle hT = arcsin(sin
(g/2)/m2) is the internal angle at which an incident light ray enters
the grain core from within the space-weathered layer. It is also the
angle at which it subsequently strikes the space-weathered layer
from within the core before leaving the grain at an exit angle of
g/2. The H-function, defined by the well-known radiative transfer
theory of Chandrasekhar, is defined as:
1
pffiffiffiffiffiffiffiffiffiffiffiffi
1
1
Hðl; wÞ ¼ 1 1 1 w l r 0 þ 1 r 0 r 0 l ln 1 þ
;
2
l
pffiffiffiffiffiffiffiffiffiffiffiffi
l ¼ cosðg=2Þ; r0 ¼ 2=ð1 þ 1 wÞ 1
ð3Þ
Fig. 1. The 2D model regolith grain.
Thus, the regolith grain is defined in terms of five parameters.
We restricted the value of grain parameters to lie within ranges
comparable to values identified for the fine fraction of regolith
samples returned to Earth by the Apollo and Luna missions. The
parameter ranges are shown in Table 2.
158
A. Fearnside et al. / Icarus 268 (2016) 156–171
Table 1
The single scattering albedo (wx) and back-scattering activity (Xx) defined in terms of scattering and transmission activities at boundaries within the grain. The three different
media are signified by the following subscripts: ‘‘0” = vacuum; ‘‘1” = space-weathered layer; ‘‘2” = grain core. The polarisation states are denoted by the symbol x in which x = \
denotes the polarisation perpendicular to the scattering plane and x = || denotes the polarisation parallel to the scattering plane.
Boundary location
Single scattering albedo
wx ¼
Scattering (vacuum M shell)
x
s01
Transmission (vacuum M shell)
Back-scattering activity
x 1s x ea2 D þt x s x t x e2a2 D
s01
ð 21
Þ 01 21 21
x
x
x
x
X x ðg=2Þ ¼ S01
ðg=2Þ þ T 01
ðg=2ÞS21
ðhT ÞT 21
ðhT Þe2a2 D
x x a2 D
x ea2 D
t 21 e
ð1s21
Þð1t01
Þ
x
x r x e2a1 d
ð1r10
Þð1r01
Þ 12
x
¼ r 01 þ
x r x e2a1 d
1r
ð 10 12
Þ
x
x
ðg=2ÞÞ R21
ðhT Þe2a1 d
ð1R01
x
x
ðg=2ÞR12
ðhT Þe2a1 d Þ
ð1R01
x
x
ðg=2ÞÞð1R21
ðhT ÞÞea1 d
ð1R01
x
T 01
ðg=2Þ ¼ 1R
x
x
ðg=2ÞR21
ðhT Þe2a1 d Þ
ð 01
x
x
ðhT ÞÞð1R01
ðg=2ÞÞea1 d
ð1R21
x
T 21
ðhT Þ ¼ 1R
x
x
ðg=2ÞR21
ðhT Þe2a1 d Þ
ð 01
x
x
ð1r01
Þð1r12
Þea1 d
x r x e2a1 d
ð1r10
Þ
12
x
x
ð1r21
Þð1r10
Þea1 d
¼ 1r
x r x e2a1 d
ð 10 12
Þ
x
x r x e2a1 d
ð1r21
Þð1r12
Þ 10
x
¼ r 21
þ
x r x e2a1 d
1r
ð 10 12
Þ
x
t01
¼
Transmission (shell M core)
x
t21
Scattering (shell M core)
x
s21
2
x
x
¼ R01
ðg=2Þ þ
S01
x
x
ðhT Þ ¼ R21
ðhT Þ þ
S21
2
x
x
ðhT ÞÞ R01
ðg=2Þe2a1 d
ð1R21
x
x
ðg=2ÞR21
ðhT Þe2a1 d Þ
ð1R01
-5
x 10
18
-0.4
(a)
-0.5
-0.6
14
-0.8
12
-0.9
10
-1
8
Increasing D
or α, φ d
6
-1.3
-1.4
4
-1.5
-1.6
-1
0.04
-0.8
0.02
-0.9
0
-1
-0.02
-1.1
Increasing n
-1.2
-0.04
-1.3
-0.06
-1.4
-0.08
-1.5
2
-0.8
-0.6
-0.4
-0.2
0
0.2
log10 (B)
0.06
-0.7
log10 (Pmax)
log10 (Pmax)
-0.7
-1.2
(b)
-0.5
-0.6
-1.1
0.08
-0.4
16
-1.6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
-0.1
log10 (B)
Fig. 2. Umov plots calculated according to the 2D grain model. (a) The effects of a variation in grain diameter. Here, grain core diameter (D) intervals are indicated by hue. The
colour side bar indicates core diameter in metres. The same type of graphical structure is reproduced by variations in space weathering parameters (/, d) and in core
absorption coefficient (a2). (b) The effects of a variation in grain core refractive index (n2). Data of common refractive index lies within the plot along a common longitudinal
band, indicated by hue. The colour side bar indicates refractive index as a deviation relative to n2 = 1.68. (For interpretation of the references to colour in this figure legend, the
reader is referred to the web version of this article.)
With the optical wavelength and absorption coefficient fixed,
eighteen different values were used for each of the remaining four
variable grain parameters spanning each parameter range. This produced 184 different combinations of grain parameters, representing
184 different grains. We calculated the maximum degree of linear
polarisation (Pmax) of light scattered from each one of these different grains, and the brightness of light scattered at the phase angle
at which Pmax occurred. This information was then plotted upon
on a scatter graph with logarithmic axes, collectively for all of the
184 different grains, and we found that the expected linear correlation of Umov’s Law was reproduced as is shown in Fig. 2.
Fig. 2(a) and (b) shows how a variation through the range of values of each of the five grain parameters influences the structure of
the Umov plot for the model regolith grains. Fig. 2(a) shows that the
effect of increasing the value of any one of the mineral core diameter (D), the thickness (d) of the space-weathered layer, the proportion (/) of npFe present within the space-weathered layer, and the
absorption coefficient (a2) of the grain core, is to reduce the brightness of scattered light and increase the maximum polarisation. The
result is to extend the scatter of data points within the plot linearly
along the direction of correlation. This is due to the grain becoming
more opaque. Increased opacity reduces the proportion of light
emerging from within the grain following internal scattering processes that tend to de-polarise the light. However, the proportion
of relatively strongly polarised light resulting from specular surface
reflection is not reduced in this way. Consequently, the overall
degree of polarisation tends to rise as the grain parameters are varied to increase grain opacity. In contrast, as Fig. 2(b) shows, a variation in the value of the real part of the refractive index (n2) of the
mineral core material has the effect of simply shifting, wholesale,
the linear distribution of data points in a direction transverse to
the usual direction of correlation. This shift is generally uniform
along the entire length of the distribution of data points such that
a linear correlation is preserved. This is analogous to the effect
expected of the Polarimetric Anomaly Parameter, (Pmax)aA, proposed by Shkuratov and others. In the remainder of this paper, we
refer to this shift in the Umov plot as a Polarisation Anomaly to distinguish it from the Polarimetric Anomaly Parameter, and we discuss the relationship between these two quantities further. The
Polarisation Anomaly is strongly correlated to the real component
(n2) of the refractive index of the grains associated with it. An
Umov-type scatter plot may now be viewed as being composed of
a succession of adjacent, parallel bands of data as shown in Fig. 2
(b). Each data band has the same unique Polarisation Anomaly
along its length and each is populated by data points derived from
grains sharing the same refractive index (real component). However, grains populating any one band differ from each other in all
other grain parameters (D, d, /, a) over a range of values. Fig. 2(b)
illustrates this graphically by identifying each such band with a different colour denoting the refractive index. Thus, by using the
Polarisation Anomaly, a variation in grain composition can be identified independently of the otherwise confusing effects of grain size,
opacity and, most notably, space weathering.
3.2. Calibration: refractive index from polarisation anomaly
Here we use the results of the 2D grain model to derive a simple
formula for use in calculating the refractive index of a regolith
159
A. Fearnside et al. / Icarus 268 (2016) 156–171
-0.6
(a)
Small
Large
(c)
-0.7
-0.8
log10 (Pmax)
-0.9
-1
-1.1
-1.2
-1.3
-1.4
-1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
log10 (B)
-0.6
-0.6
(b)
-0.7
Low
High
2 lines
-0.8
-0.9
log10 (Pmax)
log10 (Pmax)
-0.8
-1
-1.1
-0.9
-1
-1.1
-1.2
-1.2
-1.3
-1.3
-1.4
-1.4
-1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Low
Medium
High
(d)
-0.7
2.8
-1.5
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
log10 (B)
log10 (B)
Fig. 3. (a) A small portion of a simulated regolith comprising a loose pack of grains. A single ray path incident upon (from left) and exiting from (to right) the pack is shown.
Dashed lines indicate the boundary between a host grain core and its space-weathered shell layer. (b–d) Umov plots for 144 different simulated regoliths spanning different
combinations of structural and optical parameters. (b) This Umov plot indicates data points associated with two of the three different values of refractive index of regolith
grain (relatively ‘‘low” n = 1.58, and relatively ‘‘high” n = 1.78). Confidence intervals (dashed lines) are shown for each refractive index data group. These define the 2r
confidence limits in the least-squares linear regression calculated for the group in question. (c) This Umov plot indicates data points associated with ‘‘small” (D = 2.4 lm) and
‘‘large” (D = 4.8 lm) average grain sizes. A similar effect is achieved by ‘‘small” (a = 3386 m1) or ‘‘large” (a = 5080 m1) absorption coefficient. (d) This Umov plot indicates
data points associated with ‘‘low”, ‘‘medium” and ‘‘high” values (/ = 0.005, 0.015, 0.025) of npFe content within the space-weathered layer coating grains.
Table 2
Structural and optical parameters defining the simulated regolith grain.
Grain parameter
Range
Core diameter (D)
Weathered layer thickness (d)
Refractive index (real part, n2)
Absorption coefficient (a2 = 4pk2/k)
Proportion (/) of npFe
10–200 lm
10–220 nm
1.58–1.78
5080 m1 (k = 500 nm)
0.01–0.03
according to the Polarisation Anomaly it displays on an Umov plot.
This formula is applicable to materials with refractive indices n2 in
the range: 1:5 6 n2 6 2:0. This range covers the expected range of
refractive indices of the lunar regolith. The single scattering albedo
(w) of a particle can be defined in terms of its scattering efficiency
(QS) and its extinction efficiency (QE) as w = QS/QE. For closelyspaced particles, QE 1 and so w QS. This can be expressed in
terms of the reflection coefficient Se for surface reflection of externally incident light, the reflection coefficient Si for surface reflection of internally incident light and the internal transmission
factor h for light within the particle, as QS = Se + h(1 Se)(1 Si)
(1 Sih)1. As a first approximation, the surface scattered component Se of the reflection coefficient represents wholly polarised
light whereas the internally scattered component h(1 Se)(1 Si)
(1 Sih)1 represents wholly unpolarised light, depolarised by
internal scattering processes. The degree of polarisation, P, of the
scattered light may then be reduced to P Se/QS Se/w. Taking log-
arithms of this expression gives log10(P) = log10(Se) log10(w). We
assume that the albedo (A) of a particulate surface is proportional
to the back-scattering activity of the particles forming it. A comparison of the expressions for back-scattering activity (Xx) and
the single scattering albedo (wx) of Table 1 reveals a close relationship between the two, in which the latter includes the former
together with forward scattering terms. With the benefit of a hindsight knowledge of Umov’s Law, we propose that this relationship
can be expressed in the form wx X cx2 , where c2 is a constant. This
allows us to write wx Ac2 . Our logarithmic expression then takes
the functional form of Umov’s Law: log10(P) = c1 c2log10(A) in
which c1 = log10(Se). The Polarisation Anomaly, Dpol, being the perpendicular separation between two parallel such lines, is then
obtained by:
1
Sð2Þ
Dpol ¼ log eð1Þ
b
Se
where b ¼
!
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ c22 . The terms SðjÞ
e ðj ¼ 1; 2Þ are the surface reflection
coefficients for a grain of refractive index mj. For irregular particles
with a complex refractive index mj = nj + kj with 1:2 6 nj 6 2:0 and
kj 1 following both Hapke (Hapke, 2005) and Shkuratov
(Shkuratov, 1981), we may write, approximately:
SðjÞ
e ¼ b1
ðnj 1Þ2
ðnj þ 1Þ2
þ b2
ð5Þ
160
A. Fearnside et al. / Icarus 268 (2016) 156–171
Hapke suggested that b1 = 1.0 and b2 = 0.05. However, Shkuratov suggests that surface reflection may be refined by setting
b1 ¼ b1 g; G; FF0 to be a phase coefficient of surface porosity, connected with the coefficient of volumetric porosity. The term F represents the total area of microsurfaces of a regolith specularly
reflecting towards the observer from a given region of area F0.
The term G allows for the departure of the Fresnel reflection indicatrix from a d-function due to micro-roughness at grain surfaces.
While we do not implement such detailed methods here, we do
generalise Eq. (5) such that b1 and b2 are free parameters. By solving Eq. (5) for n2 (i.e. j = 2) and substituting for Sð2Þ
using Eq. (4),
e
one may write an expression for the refractive index of grains in
terms of their Polarisation Anomaly (Dpol) relative to an Umov scat^2 :
ter line for other grains of a reference refractive index n
1=2
bDpol
b1 b2 þSð1Þ
þ2b3 b1 Seð1Þ 10bDpol b2
e 10
^2 Þ ¼
;
n2 ðDpol ;c2 ; n
b1 þb2 Seð1Þ 10bDpol
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^ 2 1Þ
ðn
2b1
^ 2 Þ2 1þb3 g
b
b3 fðn
ð6Þ
Sð1Þ
þb2 þ
3
e ¼ b1 ^
ðn2 þ1Þ
^ 2 þ1Þ2
ðn
Here we have generalised the result by adding a free parameter b3.
When b3 = 1 we find Seð1Þ to be defined as Eq. (5), as expected. In
order to derive values of b1, b2 and b3 we calculated Dpol for many
values of n2 using our 2D model, and then fitted Eq. (6) to the
results. The values of b1, b2 and b3 were adjusted to optimise this
fit. We found that a good fit only occurs when b3 0 when applied
to the range of n2 being considered. We found excellent agreement
when n2 was restricted to the range 1:5 6 n2 6 2:0, with
b1 = 0.9568, b2 = 0.0178 and b3 = 0. Eq. (6) then takes the succinct
form:
bDpol
b1 b2 þ Sð1Þ
e 10
^2 Þ ¼ ;
n2 ðDpol ; c2 ; n
bDpol
b1 þ b2 Sð1Þ
e 10
Sð1Þ
e ¼ b1
^ 2 1Þ
ðn
þ b2
^ 2 þ 1Þ
ðn
ð7Þ
^ 2 associated
With knowledge of a reference refractive index n
with a reference Umov line of measured gradient c2, one can use
^ 2 Þ of grains associEq. (7) to derive the refractive index n2 ðDpol ; c2 ; n
ated with a parallel, second Umov line offset from the first by a
Polarisation Anomaly Dpol. This method will be used extensively
in Sections 4 and 5 of this paper to derive the refractive index of
granular terrestrial and lunar materials. Fig. 5 shows the curve: n2(Dpol) calculated using the 2D grain model for a reference refractive
^ 2 ¼ 1:526. Together with this curve is shown a second
index of n
curve (solid line) corresponding to Eq. (7).
3.3. Three-dimensional (3D) model
The 2D grain model is, of course, limited in its scope not merely
because it is only 2D but also because it represents a grain in isolation. In order to overcome these limitations we developed a more
realistic 3D multi-grain model of the lunar regolith which does
include the effects of grain-to-grain scattering of light.
3.3.1. Model basics
Each of the grains in a given multi-grain model regolith was
composed of an irregularly-shaped polygonal core of uniform mineral material coated with a uniform space-weathered layer. The
refractive indices and absorption coefficients of these polygonal
grains were calculated in the same way as was done for the grain
of the 2D model. We employed geometrical optics to trace the
paths of many light rays propagating through a collection of many
such irregular grains representing a regolith. In doing so, we computed the phase curves for the brightness and for the degree of linear polarisation of light scattered by many different types of such
regoliths each distinguished by the average size and/or composition of their grains. The state of polarisation of each light ray was
represented by its Stokes vector, which was calculated using the
methods described by Muinonen et al. (1996). We define the
Stokes vector of a ray as M = [I, Q, U, V] in general. Rays of light initially incident upon the regolith are un-polarised so that MI =
[1, 0, 0, 0]. At each intercept between a light ray and a regolith grain
boundary, the Stokes vectors for the scattered and transmitted
components of light rays are calculated using MS = R K MI and
MT = T K MI respectively. The matrices R, T and K are the Mueller matrices for reflection, for transmission, and a rotation matrix
K = K(u):
2
1
0
0
0
0
0
0
1
3
6 0 cosð2uÞ sinð2uÞ 0 7
7
6
KðuÞ ¼ 6
7
4 0 sinð2uÞ cosð2uÞ 0 5
The rotation matrix allows for a change of scattering plane at
each intercept. All rays are initially constrained to a single 2D plane
whereby the rotation matrix applies a rotation angle (u) of zero
radians. Fig. 3 shows a small part of a typical 2D pack of grains
in a simulated regolith, and the 2D path of a single ray which
enters the regolith at an incidence angle of 80°. Eventually it
escapes back to space having passed through many grains. To
approximate the 3D scattering of the light rays, a random rotation
ui was applied to each one of j line segments of a light ray between
successive grain boundary intercepts, using rotation matrix K(ui),
(i = 1, 2, . . .j), piece-wise for the full 2D path taken by a light ray.
This technique thereby applied j successive rotations to the segments of the 2D path of a light ray from incidence into the regolith
to emergence from it. This enforced a change of scattering plane at
each intercept of the ray with a grain boundary while preserving
the Fresnel amplitudes previously calculated along the 2D path.
The random rotation angles ui were applied up to 20 times
(j = 20) along the path segments of each emergent ray path so that
a single emergent ray produced 20 distinct 3D ray paths. The phase
angle (g) for each 3D ray path was defined as the angle between the
incoming and emergent ray. Emergent rays were grouped into bins
according to phase angle, with each bin covering 1° of phase angle
across the range from 0° to 160°. An average Stokes vector M = [I,
Q, U, V] for all the rays in each bin was calculated to determine the
average intensity and polarisation for that range of phase angle.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
The linear polarisation was defined by P ¼ Q 2 þ U 2 =I. The irregular polygonal shapes of the grains within a model regolith were
varied randomly and a minimum 0.1 lm gap was imposed
between grain vertices, whereas other physical properties were
kept constant for a given regolith simulation. Thus, all grains in a
given simulated regolith shared the same thickness of spaceweathered outer layer, npFe concentration, average grain size
and core refractive index. The upper layers of the lunar regolith
are believed to be very loosely arranged and form a so-called
‘fairy-castle’ structure. Accordingly, a fraction of grains were randomly deleted from the grain pack in order to simulate an open
structure. Relatively smaller grains populated the upper layers of
the model regolith as is expected of the lunar regolith (Papike
et al., 1981). Those smaller grains are expected to dominate the
optical properties of the regolith surface. Indeed, our simulations
confirm that incoming light is quickly absorbed and tends not to
penetrate far into the regolith. We simulated 4000 distinct regoliths, and traced 360,000 input light rays through each one of those
regoliths. Each simulation started by illuminating a simulated
A. Fearnside et al. / Icarus 268 (2016) 156–171
regolith with a set of 10 un-polarised light rays. The paths of the
light rays were traced through the regolith until they either
emerged from its illuminated surface or were attenuated to an
intensity below 1% of their initial intensity. The process was
repeated for each new incidence angle up to incidence angles of
80°. To investigate the effect of different grain sizes and properties,
144 different regoliths were simulated representing all of the different combinations of the parameters listed in Table 3.
3.3.2. Umov’s Law
Fig. 3 shows three Umov plots derived from our 3D model. In
each of Fig. 3(b)–(d), we indicate how different data points are
associated with different grain parameters. In particular, Fig. 3(b)
shows different data points according to the refractive index of
the grain cores of the simulated regolith. It is clear that the 3D
model has reproduced the banding structure predicted by the 2D
model discussed above. It shows that Polarisation Anomaly is governed by refractive index differences. A variation in other grain
parameters of the simulated regolith do not produce this banding
pattern. For example, increasing either the average grain size or
the npFe content of grain coatings simply moves data points along
(to the upper left) the Umov correlation line, not across it. This can
be seen in Fig. 3(c) and (d) and is also predicted by our 2D model.
3.3.3. Linear regression and error analysis
We have calculated regression lines for each of the two linear
groups of data points presented in Fig. 3(b). These are indicated
in Fig. 3(b) in terms of their associated confidence intervals. The
confidence intervals appear as pairs of lines bounding an associated linear data group. They define a 2r interval of confidence in
the least-squares regression line calculated for the data group in
question. One can see that the confidence intervals for the two data
groups do not overlap. This indicates a significant level of confidence that the single refractive index value assigned to each linear
data group can be well represented by a simple linear regression
line. While our error analysis employs 1r confidence intervals,
these confidence intervals are too small to be visible on the scale
of Fig. 3(b) and so we present 2r confidence intervals there for
illustrative purposes only. A 1r confidence interval in a linear
regression upon either data group of Fig. 3(b) corresponds to an
interval in refractive index, as measured from Polarisation Anomaly, of about ±0.0026.
3.3.4. Conclusions
The 3D regolith model confirms the relationship between Polarisation Anomaly and refractive index predicted by the simple 2D
grain model. Both confirm that Polarisation Anomaly is independent of the effects of grain size variation. This appears to be contrary to the interpretations of Shkuratov and of Dollfus regarding
Umov’s Law and the ‘Polarimetric Anomaly Parameter’. Recall that
‘polarisation excess’, ‘polarisation deficit’ and the ‘Polarimetric
Anomaly Parameter’ that quantifies them, continue to be interpreted as a measure of variations in the median value of regolith
Table 3
Structural and optical properties defining simulated regolith grains. The three
absorbtion coefficient values (a = 4pk/k) correspond to wavelengths of 500 lm,
600 lm and 750 lm. A total of 144 different combinations of the above parameter
values were used.
Grain property
Value
Mean grain size (lm)
Minimum grain separation (lm)
Weathered layer thickness (nm)
Refractive index (real part, n)
Absorption coefficient (a = 4pk/k, m1)
Proportion (/) of npFe
0.6, 1.2, 2.4, 4.8
0.1
100
1.58, 1,68, 1.78
5080, 4233, 3386
0.005, 0.01, 0.015, 0.025
161
grain size. We now propose a new interpretation. That is, these
effects are all caused by variation in the bulk refractive index of
the lunar regions being observed. Regions showing a ‘polarisation
deficit’ (Aristarchus Plateau, Marius Hills) do so because they comprise regolith grains having notably lower refractive index. Similarly, regions showing a ‘polarisation excess’ do so because they
comprise regolith grains having notably higher refractive index.
We shall discuss both matters further below with reference to
new polarimetric measurements of the Moon. However, before
that, we discuss a direct experimental test of our predictions
regarding Polarisation Anomaly.
4. Experiment
While two separate models each predicted that Polarisation
Anomaly is caused by variation in the refractive index of regolith
grains, we resolved to test these theories directly against
experiments.
4.1. Goniometer measurements
Using a goniometer, polarisation phase curves were determined
for each of 17 powdered samples of three different materials of
known refractive index (real component, n2). In the goniometer a
monochromatic video camera (model DMK21AU04.AS) with
focussing lens, was mounted at a fixed angular position relative
to each sample. A white LED lamp mounted at the far end of a
rotating arm illuminated the sample from a fixed distance at variable angles of illumination. The angle subtended by the lamp arm
and the line-of-sight of the video camera defined the phase angle
(g) of the illuminated sample. Samples were imaged through a linear polarising filter and a green pass-band filter, having an 80 nm
wide pass band centred upon 540 nm. The lamp emissions were
found to be un-polarised. A set of still images of each sample
was recorded when illuminated by the lamp at a common specified
phase angle. Each still image within a set corresponded to a different angular position of the linear polarising filter. The sample
brightness (I) at a given phase angle was defined as the average
pixel value within a sample image. This changed with changing
polarising filter angular position (h), conforming very well to
Malus’ Law which takes the form I = Acos2(h + u) + B, in which A
and B are constant and u is an arbitrary phase angle. From this
the degree of linear polarisation was calculated as P = A/(A + 2B).
For each sample material, a sequence of such polarisation measurements was made at different phase angles spanning a phase
curve, and from that phase curve the maximum degree of polarisation (Pmax) was determined for that sample. The brightness of a
given sample was determined as the measured reflection from
the sample at the minimum possible phase angle of the goniometer. This minimum phase was the same in all cases and was chosen
to minimise the effects of grain shadowing within a sample. The
three sample materials were black soda lime glass, silicon carbide
and JSC-1a lunar regolith (mare) simulant (Joy, 2014). Each sample
material was sieved to separate the grains into different size ranges
as listed in Table 4. Refractive index values for the soda lime glass
and the silicon carbide samples were obtained from the literature
(Rubin, 1985; Shaffer, 1971), whereas the refractive index for JSC1a was directly measured using the Becke line method (Bowles,
2014). Each such value of refractive index is in respect of green
light of wavelength 540 nm.
While silicon carbide and soda lime glass are not especially
lunar-like, they have been chosen because of their uniform composition, known chemistry and widely differing refractive indices relative to each other. They provide controllable end-member
materials with which to test our theoretical models which should
162
A. Fearnside et al. / Icarus 268 (2016) 156–171
Table 4
Sample grain size ranges.
Material
n2
1st size
2nd size
3rd size
4th size
5th size
6th size
Soda lime glass
1.526
455–355 lm
300–250 lm
Silicon carbide
2.671
F80 (165 lma)
F220 (63 lma)
JSC-1a
1.603
355–300 lm
180–150 lm
125–90 lm
7th size
212–180 lm
180–150 lm
150–120 lm
125–90 lm
90–75 lm
F400 (22 lma)
F600 (15 lma)
–
–
–
90–75 lm
75–45 lmb
<45 lmb
–
a
Note: ‘‘F” values for silicon carbide are according to U.S. Department of Commerce Commercial Standard CS 271-65 for which mean, maximum and minimum particle
diameters (D) are as follows. F80: Dmean = 165 lm, Dmax = 292 lm, Dmin = 102 lm. F220: Dmean = 63 lm, Dmax = 102 lm, Dmin = 20 lm. F400: Dmean = 22 lm, Dmax = 45 lm,
Dmin = 11 lm. F600: Dmean = 15 lm, Dmax = 35 lm, Dmin = 9 lm.
b
Note: Refractive index of these grains possibly less than 1.603 due to differing composition. Not confirmed.
apply to any granular material, not just lunar regolith. Of course,
JSC-1a is widely regarded as being a good regolith simulant and
is included for comparison. However, it is not clear that the composition (and therefore refractive index) of JSC-1a is uniform as a
function of grain size. Below we suggest that it may vary.
4.2. Umov’s Law
The value of maximum polarisation and sample brightness was
determined for each sample, and the result is plotted as an Umov
plot in Fig. 4.
Fig. 4 clearly shows the linear trend line associated with Umov’s
Law. Regression lines are shown for the data associated with each
of the three materials of Table 4. The regression lines for the soda
lime glass and for the silicon carbide are separated and approximately parallel. In both of these samples, the data associated with
the largest grains occupy the upper left parts of the regression line
(lesser brightness, greater polarisation) whereas the data associated with ever smaller grains occupies increasingly the lower right
portions (greater brightness, lesser polarisation). These features of
the data are fully in accordance with both the 2D and the 3D regolith models (see Figs. 2 and 3). Most notably, the measured transverse separation between the regression lines for soda lime glass
and silicon carbide has a mean value of 0.288. With refractive
^ 2 ¼ 1:526 for soda lime glass and n2 = 2.671 for silicon
indices of n
carbide, at a wavelength of light of 540 nm, the difference between
these two refractive indices is 1.145. Our 2D grain model predicts
that such a refractive index difference should manifest itself in the
form of a Polarisation Anomaly of Dpol = 0.28. This is in close agree-
Fig. 4. Umov plot for peak polarisation vs. brightness of all samples of soda-lime
glass, silicon carbide and JSC-1a lunar simulant samples of all grain size ranges.
Confidence intervals (1r) are indicated for the silicon carbide and soda lime glass
data (dashed lines) and the corresponding upper and lower extremes (Dmax, Dmin)
of the measurement error interval in Polarisation Anomaly (Dpol).
ment with the measured value of Dpol = 0.288. Furthermore, the
measured transverse separation between the regression line for
soda lime glass and the cluster of data points for the largest four
samples of JSC-1a, is Dpol = 0.048. The 2D grain model predicts that
this Polarisation Anomaly arises from a refractive index for JSC-1a
of n2 = 1.59. This is very close to the independently-measured
(Bowles, 2014) refractive index for JSC-1a of n2 = 1.603 for light
of wavelength 540 nm. Table 5 summarises the comparisons
between actual refractive indices and the predictions of the 2D
grain model as applied to our goniometer measurements.
Chemical analysis of JSC-1a (Orbitec, 2007) reveals that its main
components are plagioclase feldspar and basaltic glass. The plagioclase component has a higher proportion of Al2O3 and a lower proportion of FeO2 than has the basaltic glass component. This will
directly influence the refractive indices of these components. We
suspect that the smallest grain-sized samples of JSC-1a in our
experiments may have been dominated by lower-index plagioclase
grains, whereas the larger grain-sized samples may have been
dominated by higher-index basaltic glasses. The milling process
by which JSC-1a is prepared is likely to preferentially enrich its finest fraction with grains of the relatively more friable plagioclase.
This could explain why the samples of JSC-1a having smaller grains
appear to have a refractive index comparable to that of our soda
lime glass samples. Microscopic inspection of the sample grains
revealed that the larger grain sizes contained an elevated proportion of highly vesiculated glass fragments not present in the smaller grain-sized samples. This vesicular structure is likely to be the
cause of the clustering of data points associated with the larger
grain-size samples in the upper left end of the Umov scatter plot
for JSC-1a in Fig. 4. This is because, while the larger grain sizes
may possess greater size, they also may possess greater voidage/
vesicles which reduce the effective optical depth/size of those
grains, and also increase the probability of outward scattering of
light from a grain before that light has fully traversed the grain.
The effect may be to make the grains appear optically smaller than
they are physically.
Fig. 5 shows how the magnitude of the Polarisation Anomaly
corresponds with a variation in the real part of the refractive index
of a grain modelled according to the 2D and 3D regolith models
and measured experimentally (Table 5) for soda lime glass, JSC1a and silicon carbide. The agreement is encouraging, especially
across the range 1:5 6 n2 6 2:0. Here, we have chosen a value for
^ 2 ¼ 1:526 from which to deterthe reference refractive index of n
mine Polarisation Anomaly and this is representative of the refrac-
Table 5
Measured refractive index values and actual values (⁄measured according to the
Becke line method).
Material
Dpol
n2 (predicted)
Error interval
n2 (actual)
SiC
JSC-1a
0.288 ± 0.015
0.048 ± 0.015
2.74
1.593
2.62–2.94
1.57–1.61
2.671
1.603 ± 0.01⁄
A. Fearnside et al. / Icarus 268 (2016) 156–171
3
2.8
REFRACTIVE INDEX
2.6
2.4
2.2
2
1.8
1.6
1.4
0
0.05
0.1
0.15
0.2
0.25
0.3
POLARISATION ANOMALY
Fig. 5. Calibration curves for predicting refractive index values n2 derived from
^ 2 ¼ 1:526. (1) The 2D
Polarisation Anomaly relative to a reference refractive index n
grain model (dashed line); (2) the analytical Eq. (7) (solid line); (3) the 3D grain
model (triangles); (4) measured Polarisation Anomaly for soda lime glass, JSC-1a
and silicon carbide (solid dots) at a wavelength of 540 nm.
tive index of the soda lime glass powder employed in the present
experiments. This is also, of course, close to the value of the refractive index of plagioclase which is abundant in lunar highland terrains. Terrestrial samples of plagioclase are known to have
refractive indices which range in value from about 1.485 to 1.60
depending on composition and structure (e.g. fused, thetomorphic
or crystalline). Pyroxenes typically range from 1.591 to 1.755 in
refractive index. For olivines, the range is from about 1.651 to
1.869, while lunar ilmenite samples are known to have a refractive
index (real part) of about 2.58. These values are found from a
review of the literature shown in more detail in Table 6, and correspond to optical wavelengths of 550 nm. We use these ranges as a
guideline for the likely values of refractive index we might expect
to see in lunar regoliths of different mineralogy. This helps us to
apply our predictions regarding Polarisation Anomaly to remote
observations of the Moon, as we discuss in the following sections
of this paper. Preliminary measurements applied to controlled
mixtures of silicon carbide grains and soda lime glass grains
(refractive index end members) indicate that the Polarisation Anomaly associated with the mixture changes in proportion to the relative contribution of the end members of the mixture. This is seen
to occur both when the end members are of similar grain size, or of
different grain size. This is in accordance with preliminary theoretical modelling and is a topic of on-going work. Its further discussion is beyond the scope of the present paper. Notably, when
controlled mixtures of grains of the same refractive index (e.g. silicon carbide alone) but different grain size are employed (grain
size end members), such a mixture produces no Polarisation Anomaly. In that case, each grain size end member is associated with a
different position along the linear trend line of the same Umov plot
and the corresponding position associated with the mixture moves
along (but not away from) that trend line, between the endmember positions, in proportion to the relative contribution of
the end members of the mixture. This is as expected from theoretical modelling described above and reinforces the conclusion that
refractive index changes induce Polarisation Anomaly, whereas
grain size changes alone do not.
163
shown in Fig. 4 for silicon carbide and soda lime glass. The number
of data points in the Umov plots for each material is limited simply
by the limited number of different sieve sizes available for separating each sample into different grain size groups and this consequently influences the size of the confidence intervals. Each
confidence interval represents the confidence bounds for the linear
regression line calculated for the data set concerned. We use these
extreme limits to estimate error margins in Polarisation Anomaly
measured using the regression lines in question. For example,
regarding silicon carbide, Fig. 4 shows how the mean value
(Dpol = 0.288), upper value (Dmax = 0.303) and lower value
(Dmin = 0.273) are measured using the mean, upper and lower confidence interval limits of the regression lines for silicon carbide and
soda lime glass. This leads to the estimated Polarisation Anomaly
value of Dpol = 0.288 ± 0.015 for silicon carbide relative to soda lime
glass. The same procedure was also applied to the Umov plot for
JSC-1a at the region of the plot associated with the largest grains.
Fig. 5 shows these error margins whereas the estimated error in
measuring the refractive index of JSC-1a (Bowles, 2014) was
±0.01 and is too small to be visible. Table 5 indicates the estimated
error values in Polarisation Anomaly for silicon carbide and JSC-1a,
and the resulting error interval in the corresponding value of
refractive index when calculated by applying the measured Polarisation Anomaly to Eq. (7).
5. Observations of the Moon
In this section we describe polarimetric observations of parts of
the lunar surface. We show how we have identified Polarisation
Anomalies within these observations and how we have converted
these Polarisation Anomalies into maps of refractive index across
the lunar surface. The method of polarimetric observation is the
same, in its basic principle, as the method described above regarding our goniometer measurements. Of course, in our lunar observations we have replaced the goniometer grain samples by the lunar
surface itself, and the imaging camera lens is replaced by a telescope as described below.
5.1. Equipment
Schmidt-Cassegrain or Maksutov-Cassegrain telescopes were
used, with mirror diameters of 20 cm or 25 cm. These axisymmetric instruments have no off-axis reflections that might
affect the measured polarisation. Monochromatic CCD video cameras of two types were used to record data. They were, the
DMK21AU04.AS camera containing a Sony ICX098BL CCD chip,
and the Lumenera Infinity-2 camera containing a Sony ICX274
CCD chip. We found that each of the CCD cameras has a sensitivity
to linearly polarised light depending on the orientation of the
polarisation axis relative to the CCD chip. The physical reason for
this sensitivity is not clear. To resolve this problem, we calibrated
our cameras against an un-polarised light source and derived a correction factor which we applied not only to the telescopic observations, but also to the goniometer measurements described above.
The polarising filters were optically-polished glass polarisers
mounted to be rotated to calibrated angular positions relative to
the telescope/camera. Video sequences were recorded in visible
light through an infra-red rejection filter with a transmission range
between 400 and 710 nm to cut out un-focussed near infra-red
light.
4.3. Error analysis
5.2. Observational technique and data acquisition
To estimate error margins we calculated 1r confidence intervals for each linear regression line shown in Fig. 4. Examples are
We recorded 21 video sequences of a selected region of the
Moon imaged telescopically through a linear polarising filter when
164
A. Fearnside et al. / Icarus 268 (2016) 156–171
Table 6
Refractive index values for terrestrial and lunar materials.
Fused
Plagioclase
a
c
d
e
f
g
h
j
k
l
m
o
p
q
s
t
u
v
w
x
y
b
Crystalline
c
Mixed grains
w
1.523 + 0.0227(An#) + 0.0264(An#)2
1.562–1.581
1.51–1.585d
–
–
1.57
1.58e
1.569–1.592f
1.535–1.60g
(Diopside)
(Pigeonite)
(Augite)
(Titanian Augite)
(Hedenbergite)
1.61–1.68h
1.683 j
1.61–1.69l
1.591–1.749f
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
1.67c
1.681–1.693k
1.685–1.703m
1.65–1.71o
1.715–1.755p
1.675–1.692q
1.684–1.714q
1.685q
1.73q
1.753q
–
–
–
–
–
–
–
–
Olivine
(Forsterite)
(Chrysolite)
(Hyalesiderite)
(Hortonolite)
(Ferrohortonolite)
(Fayalite)
1.758s
–
–
–
–
–
–
–
–
–
–
–
–
–
1.78c
1.651t
1.680t
1.733t
1.786t
1.828t
1.869t
–
–
–
–
–
–
Lunar ilmenite
Lunar simulant
Lunar pyroclast
Lunar sample
Integrated Moon
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
2.58u
1.603v
1.569x
1.67y
1.78y
Pyroxene
b
1.485–1.580
–
–
–
Thetomorphic
a
–
–
–
w
1.768 (0.118 ⁄ (Mg#))
1.726 (0.082 ⁄ (Mg#))
w
w
1. 827 (0.192 ⁄ (Mg#))
(Stoffler and Hornemann, 1972). Range given corresponds to 0 wt.% Anorthite (lower) to 100 wt.% Anorthite (upper). Wavelength is 589 nm.
(Engelhardt et al., 1970). Increasing from 0% An to 100% An.
(Hiroi et al., 2009).
(Stoffler and Hornemann, 1972). Average values for variation in anorthite content from 0% An to 100% An. Wavelength is 589 nm.
Bytownite. From Egan and Hilgemann (1979). Wavelength is 550 nm.
(Chao et al., 1971; Chao et al., 1970).
(Chayes, 1952). Anorthite content varies from 0% to 100%. Assumed wavelength is 589 nm.
(Segnit, 1953). For wt.% of Fe2O3 increasing from 0% to 20%.
(Dorschner et al., 1995) (Mg0.5Fe0.5SiO3). Generally, the formula is MgxFe1xSiO3 in which X is varied from 1.0 to 0.4, from which refractive index is (approx.) n = 1.8–0.23X.
See (h). For wt.% of TiO2 content increasing from 0% to 8%.
See (h). For wt.% of TiO2 content increasing from 0% to 8%.
See (h). For wt.% of Fe2O3 increasing from 0% to 8%.
(Binns, 1970). For (Fe + Mn)/(Fe + Mn + Mg) atom % increasing from 0% to 30%.
(Mason, 1974). Averaged over a, b and c indices.
US Geological Bulletin 1627.
(Dorschner et al., 1995) (MgFeSiO4). Generally, the formula is Mg2xFe22xSiO3 in which X is 0.5.
US Geological Bulletin 1627, for series (Mg,Fe)2SiO4.
Lunar sample 45.35.5 (Egan and Hilgemann, 1979).
Lunar regolith simulant JSC-1a (Bowles, 2014).
(Lucey, 1998; Warell and Davidsson, 2010; Carli et al., 2014). Mg# is the magnesium number Mg/(Mg + Fe), and An# is the anorthite number Ca/(Ca + Na).
(Wilcox et al., 2006). Aristarchus Plateau pyroclastic deposits.
(Hapke, 1994). Lunar Apollo sample 12070.
the lunar phase angle was between 60° and 130°. Each video
sequence was recorded at one of 21 different orientations of the
transmission axis of the polarising filter. After a video sequence
was recorded, the polarising filter was rotated (relative to the camera and Moon) to its next position, and another video sequence
recorded. This process was repeated until all 21 video sequences
were recorded within 15–20 min, to minimise the effects of the
change of illumination angle and atmospheric transparency. Video
sequences were recorded only when the sky was judged to be
transparent and when the Moon was at an altitude of at least 30°
to mitigate the effects of atmospheric turbulence and extinction
nearer the horizon. Several hundred video frames were recorded
in each video sequence, and individual frames were analysed to
assess their quality. Those of poor quality were rejected. Using
software, the remaining frames of a given video sequence were
co-aligned and stacked to produce a single stacked image of
increased signal-to-noise ratio. The stacking process also averaged
out much of the effect of any poor seeing conditions. The lunar
coordinates of the sub-solar and sub-Earth points at the time of
observation, and the pixel coordinates, latitudes and longitudes
of reference points/craters in each stacked image, were used to
convert image brightness into equigonal brightness using the
method described by Velikodsky et al. (2011). Equigonal brightness
has the effect of suppressing variations in surface brightness arising solely due to solar illumination angle above the local horizon.
5.3. Data analysis
5.3.1. Data reduction
All of the 21 stacked images were assigned a common image
coordinate system. For each pixel in this common coordinate system the variation in pixel value through each of the 21 stacked
images (i.e. the 21 polarising filter angular positions) was fitted
to Malus’ Law as described above in relation to our goniometer
experiments. The degree of linear polarisation (P) associated with
each stacked image pixel was individually calculated as P = A/(A
+ 2B) using that law. An equigonal brightness map was converted
into a polarisation map by treating each image pixel in this way.
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A. Fearnside et al. / Icarus 268 (2016) 156–171
5.3.2. Error analysis
The accuracy of our measurements of the degree of linear polarisation rests upon the confidence with which we can say that the
variation of image pixel brightness (I) follows Malus’ Law:
I = Acos2(h + u) + B, as the polarising filter angular position (h)
changes. A least-squares fit of Malus’ Law to observational data
(I, h), requires optimisation of the fitting parameters A, B and u.
This is achieved by solving the matrix equation: I = M [h] for
the optimal least-squares value of the vector M given the input
vector of equigonal brightness values: I = [I1, I2, . . .In] for a given
image pixel location across the full sequence of n different polarising filter angular positions (h). Here, n = 21. The matrix [h] contains
the position angle data:
2
cosð2h1 Þ cosð2hn Þ
6
½h ¼ 4 sinð2h1 Þ
1
3
7
sinð2hn Þ 5
1
The vector M contains all of the variable fitting parameters A, B
and u and is given by:
M ¼ ½M1 ; M 2 ; M 3 ¼
A
A
A
;
; þB
2 cosð2uÞ 2 sinð2uÞ 2
Solving for the vector M to obtain the optimal least-squares values for each of its vector components gives fitting parameter values as:
1
2
u ¼ arctan M2
2M 1
A
; A¼
; B ¼ M3 2
M1
cosð2uÞ
Upper and lower bounding values are calculated for each of
these fitting parameters which correspond with a 1r confidence
level. We then identify the combination of these parameters
required to give the maximum deviation of resulting polarisation
value P = A/(A + 2B), relative to the optimal value. This range in
polarisation deviation is used as our 1r confidence interval for
polarisation measurements. Fig. 6(a) shows an example of a data
set of 21 (n = 21) brightness values corresponding to an image
pixel position corresponding to a location in Oceanus Procellarum
west of the Aristarchus Plateau.
The optimal least-squares fit to Malus’ Law is shown together
with upper and lower bounding values according to a 1r confidence interval. The mean error as a fraction of amplitude of the fitted curve is 0.015. That curve and mean error correspond to an
(a)
estimated polarisation of 0.165 ± 0.0034 which ultimately results
in a measured refractive index of 1.75 ± 0.01.
To make a final estimate of error margins for measured refractive index values we take account of the effect of the confidence
interval we have calculated for the theoretical Umov plots we have
generated using our 3D grain model (see Fig. 3(b)). This particular
confidence interval quantifies the level of confidence we can have
in asserting that the data points within an Umov plot do indeed
correlate to a given linear correlation trend line. In Section 5.4 of
this paper we describe how a linear correlation trend line is
applied to observational data in an Umov plot in order to measure
refractive index and generate refractive index maps for the lunar
surface. In doing this we assume that the confidence intervals calculated for our 3D grain model apply equally to observational data.
When this is combined with error estimates for the polarisation
values calculated using Malus’ Law, as described above, we find
that error margins in measured refractive index values for the
lunar surface range from about ±0.01 to about ±0.02, with the typical error margin being ±0.015.
Fig. 6(b) illustrates the distribution in the size (dn) of the error
margin (±½dn) for all points in the refractive index map of Fig. 8(a)
covering northern Oceanus Procellarum including the Aristarchus
Plateau. This general distribution of errors in the refractive index
measurements is typical of each of the refractive index maps
shown in this paper. As a general rule, slightly higher error margins
of about ±0.02 in refractive index occur in the brightest regions of
the lunar surface within the mapped region, where polarisation
values are lowest. The lowest error margins, of about ±0.01 in
refractive index occur at the darkest regions where polarisation
is highest.
In the next section of this paper, we present polarimetric data
obtained in this way for regions of the Moon found by Shkuratov
to produce a ‘polarisation deficit’, as well as regions showing a ‘polarisation excess’. We apply our interpretation of Polarisation Anomaly in Umov’s Law to generate maps of the refractive index of the
regolith of these regions. We discuss how these refractive index
maps are consistent with independent estimates of the mineralogy
of these regions.
5.4. Observations and interpretation
Fig. 7 shows the results of lunar polarimetry in terms of Umov
plots for data associated with the following three regions of the
Moon: Aristarchus Plateau, Harbinger Mountains and adjacent
(b)
Fig. 6. Curve fitting of Malus’ Law to observational data. (a) Observational data (data points) of equigonal brightness of a location on the lunar surface for 21 successive
polaroid angular positions, and the optimal least-squares fitted curve (solid line) together with 1r upper and lower confidence intervals (dashed lines). (b) Distribution in the
size (dn) of the error margin (±½dn) for all points in the refractive index map of Fig. 8(a) covering northern Oceanus Procellarum including the Aristarchus Plateau.
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A. Fearnside et al. / Icarus 268 (2016) 156–171
1.6
-0.7
(a)
(d)
1.4
log10 (P)
-0.8
1.2
1
-0.9
0.8
0.6
-1
0.4
-1.1
0.2
4.4
4.5
4.6
4.7
0
4.8
log10 (B)
(b)
-0.7
(e)
2
-0.8
log10 (P)
1.5
-0.9
-1
1
-1.1
0.5
-1.2
4.4
4.5
4.6
4.7
4.8
4.9
5
0
log10 (B)
2.5
(c)
-0.8
(f)
2
log10 (P)
-0.9
1.5
-1
-1.1
1
-1.2
0.5
-1.3
4.4
4.6
4.8
5
0
log10 (B)
Fig. 7. Regions of interest (left images) and corresponding Umov plots (right images). The regions of interest are: (a) Aristarchus Plateau, Harbinger Mountains and adjacent
mare. (b) The Marius Hills and surrounding mare. (c) Mare Crisium and surrounding terrain. The colour side bar shows data density on a log scale and each Umov plot
^ 2 ¼ 1:68 ((d) and (e)) or n
^ 2 ¼ 1:60 (f). (For interpretation of the references to colour in
includes a reference line assumed to be associated with material of refractive index n
this figure legend, the reader is referred to the web version of this article.)
mare (Fig. 7(a) and (d)); the Marius Hills and surrounding mare
(Fig. 7(b) and (e)); Mare Crisium and surrounding terrain (Fig. 7
(c) and (f)). In each of Fig. 7(a)–(c) a specific region, or regions, of
interest (ROI) is indicated by a white box. As other workers have
noted, Umov’s Law is obeyed well not only by the peak value of
polarisation (Pmax) but also by polarisation values at lunar phases
close to the phase angle at which Pmax occurs. This is the case for
the results shown in Fig. 7. The value of polarisation and the corresponding equigonal brightness, are calculated for each image pixel
within a given ROI, and this information is plotted in the adjacent
Umov plot of Fig. 7(d)–(f), respectively.
The Umov plot of Fig. 7(d) corresponding to Aristarchus Plateau,
the Harbinger Mountains and mare south of the Plateau shows two
well-defined parallel linear data correlations. Compare and contrast this basic structure to the prediction shown in Fig. 3(b). Each
correlation takes the form log10(P) = c1 0.97log10(B) as indicated
by a reference line in the Umov plot where the parameters P and
B are the degree of polarisation and the equigonal brightness,
respectively. The term c1 is a constant which differs as between
the upper and lower data correlations. The upper correlation, along
which the indicated reference line passes, is associated with the
mare regions of both ROIs, whereas the lower correlation is associated with Aristarchus Plateau and the Harbinger Mountains. We
now derive a refractive index map using this Umov plot as follows.
^ 2 Þ) the measured common graWe apply to Eq. (7) (i.e. n2 ðDpol ; c2 ; n
dient (c2 = 0.97), the assumed value of reference refractive index
^ 2 ¼ 1:68Þ for the Umov data falling upon the reference line, and
ðn
the measured Polarisation Anomaly of each data point in the Umov
plot relative to the reference line. From this we calculate the refrac^ 2 Þ for the region as a whole as shown in Fig. 8
tive index n2 ðDpol ; c2 ; n
(a).
One can see that the refractive index of Aristarchus Plateau and
the Harbinger Mountains is significantly lower than that associated
with the embaying mare regions and takes values typically in
the range 1:55 6 n2 6 1:6. This is consistent with independent
estimates (Wilcox et al., 2006) which suggest that n2 = 1.569 for
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A. Fearnside et al. / Icarus 268 (2016) 156–171
(a)
(b)
1.75
1.7
1.65
1.6
1.55
1.5
1.45
1.4
Fig. 8. Refractive index map of (a) Aristarchus Plateau, Harbinger Mountains and surrounding mare; (b) Marius Hills, and surrounding mare and western highlands. Map
coverage corresponds directly to Fig. 7(a) and (b). The colour side bar shows refractive index on a linear scale. Refractive index is derived from the Polarisation Anomaly
^ 2 ¼ 1:68. (For interpretation of the references to
within the Umov plot of Fig. 7(d) or (e), relative to a reference line assumed to be associated with material of refractive index n
colour in this figure legend, the reader is referred to the web version of this article.)
pyroclastic glasses assumed to blanket these regions. Also, we note how
highland terrains on the western limb are of low refractive index
(1:50 6 n2 6 1:55) consistent with plagioclase, whereas the mare
regions have higher refractive index (1:65 6 n2 6 1:80) consistent
with pyroxenes and olivines (Shkuratov et al., 2005). Fig. 7(b)
shows an ROI encompassing Marius crater, the Marius Hills, the
surrounding mare, and the highland terrains extending to the
western limb of the Moon. The Umov plot derived from this ROI
is shown in Fig. 7(e). It shows two well-defined linear correlations
each parallel to the reference line: log10(P) = c1 1.18log10(B). The
data defining the upper correlation is associated with the embaying mare regions of this area, whereas the data defining the lower
correlation is associated with the Marius Hills and the highland
terrains. As before, we apply to Eq. (7) the measured common gradient (c2 = 1.18), the assumed value of reference refractive index
^ 2 ¼ 1:68Þ for the data falling upon the reference line, and the
ðn
measured Polarisation Anomaly of each data point in the Umov
plot relative to the reference line. The refractive index map for
the region as a whole as shown in Fig. 8(b). One can see that the
refractive index of the Marius Hills is significantly lower than that
associated with the embaying mare regions and the floor of Marius
crater. This region comprises materials of refractive index typically
in the range 1:5 6 n2 6 1:6.
Fig. 9 shows an enhanced version of the refractive index map of
Fig. 8(b) projected onto a surface relief map of the terrain of the
Marius Hills. Local surface slope (illumination angle) variations
are calculated using lunar surface topographic data made publicly
available from NASA’s Lunar Orbiter Laser Altimeter (LOLA). A
more accurate equigonal brightness of the Marius Hills was
calculated accordingly. This was used in generating an associated
Umov plot (see Fig. 7(e)). The Polarisation Anomaly and resulting
refractive indices across the Marius Hills are consequently more
accurate.
Two independent studies based upon reflection spectra of this
region have both indicated that it incorporates elevated levels of
anorthositic materials (e.g. plagioclase) (Lehman et al., 2013) or
pyroxenes of high calcium content (Besse et al., 2011). Either material can be expected to possess a reduced refractive index as compared to embaying mafic mare basalts in this region. The refractive
index map of Fig. 9 is consistent with these studies in that regard.
We note that a clear boundary is visible at the northern and eastern edges to the Marius Hills complex at which the refractive index
of the regolith falls from relatively high values of about 1.65–1.80
in the embaying mare, down to relatively low values predominantly in the range 1.55–1.60 on the complex itself. This boundary
coincides well with the geographical limits of the plateau
1.75
1.7
1.65
Marius
1.6
1.55
1.5
1.45
1.4
Fig. 9. The refractive index map of the Marius Hills and immediately surrounding mare, corresponding to Fig. 8(b). The local terrain relief is shown (with vertical scale
exaggerated) and has been taken into account when calculating equigonal brightness for the Umov plot used to measure refractive index from Polarisation Anomaly. The
colour side bar shows the refractive index linear scale. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this
article.)
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A. Fearnside et al. / Icarus 268 (2016) 156–171
containing the Marius Hills complex as defined in Fig. 5 of Besse
(Besse et al., 2011) who have studied the compositional variability
of this region in detail using data collected by the Moon Mineralogy
Mapper (M3). Fig. 6 of Besse presents a map of multi-spectral
reflectance for the region which indicates that the materials of
the plateau possess a stronger 950 nm absorption band than do
the embaying basalt flows of Oceanus Procellarum. This was
interpreted as possible evidence of high-calcium pyroxene and/or
variation in plagioclase content within the plateau.
Referring to Fig. 9 of the present paper, we note that refractive
index values of 1.65–1.80 outside the plateau coincide with those
one may expect of pyroxenes and olivines, whereas refractive
indices within the plateau as low as 1.55 are possibly too low to
be associated with pyroxenes, even those of high calcium content
such as diopside (see Table 6). One might associate such low
refractive index values with plagioclase. It is interesting to note
that the lower refractive index values seen on the Marius Hills
complex are generally associated with the locations of volcanic
domes, especially those to the south-west and north of Marius crater, which may be the source of felsic basalts according to some
current petrogenic theories.
It is worth comparing the refractive index maps of Fig. 8 with
the map of ‘Polarimetric Anomaly Parameter’ presented in Fig. 6
of Shkuratov et al. (2007). It is straight-forward to show that Eq.
(7) above may be re-cast in terms of variation in the ‘Polarimetric
Anomaly Parameter’ (Pmax)aA = d in place of the Polarisation Anomaly, Dpol, to permit a refractive index map to be recovered from
a map of ‘Polarimetric Anomaly Parameter’ as follows.
b1 b2 þ Seð1Þ R1=a
^ 1Þ
ðn
dð2Þ
^2 Þ ¼ ; Seð1Þ ¼ b1 2
n2 ðR; a; n
þ b2 ; R ¼ ð1Þ
ð1Þ 1=a
^
ðn2 þ 1Þ
d
b1 þ b2 Se R
ð8Þ
Here, d(1) is a reference ‘Polarimetric Anomaly Parameter’ associated
^ 2 , and
with regolith of an assumed reference refractive index value n
d(2) is a value of ‘Polarimetric Anomaly Parameter’ that deviates
from the reference value d(1) and is associated with a regolith of
refractive index value n(2).
A similar pattern is seen in the Umov plot for Mare Crisium.
Fig. 7(c) and (f) shows the ROI of Mare Crisium and its associated
Umov plot. The data correlation is split into two separate regions
extending parallel to an indicated reference line defined by
log10(P) = c1 0.79(B). The grouping of data lying on the trend
line, to the upper left of the plot, is associated with the surface
(a)
of Mare Crisium itself. The grouping of data points below the
trend line, to the lower right of the plot, is associated with the
terrains surrounding the mare. The refractive index map of
Fig. 10(a) was obtained by applying Eq. (7) to calculate the refrac^ 2 Þ for the region as a whole. Note that a
tive index n2 ðDpol ; c2 ; n
^ 2 ¼ 1:60 was assumed for the average refractive index
value of n
of the surface of Mare Crisium, and this lower reference value
was used due to the relatively high Al/Si concentration ratios
across Mare Crisium (0.35–0.45) as compared to other maria. This
is indicative of a less mafic surface composition in Mare Crisium
(Taylor, 1975).
However, the composition of the basalts of Mare Crisium is
more mafic than the materials of the surrounding terrains which
would tend to contain relatively more plagioclase of lower refractive index (see Table 4). This is reflected in the refractive index
maps of Fig. 10. We also see a slightly elevated refractive index
across Palus Somni, immediately west of Mare Crisium, as compared to highland terrains east and south of Mare Crisium. This
is consistent with previous works which suggest that the plains
north of crater Taruntius, and in Palus Somni, may be volcanic
units emplaced as fluid flows with a surface composition intermediate between mare basalts and highland materials. Schonfeld
(1981) has used Mg/Al ratio maps to show that this region has a
Mg/Al ratio similar to that found in mare regions (Hawke et al.,
1985). A higher Mg/Al ratio should be expected to be associated
with a higher magnesium number and, therefore, a higher refractive index. In Fig. 10(a) the floors of Picard and Peirce craters show
notably lower refractive index. Conversely, a region of elevated
refractive index appears between Picard crater and Curtis crater
to the east of Picard. Earlier studies (Andre et al., 1978) showed
that this region comprises basalts of elevated Mg/Si ratio presumed
to come from more mafic material underlying the surface of Crisium, and exhumed by the impact that formed Picard crater. Curtis
crater itself defines a striking region of unusually high refractive
index (Lat. = 14.57°, Long. = 56.79°) in Fig. 10(a). The circularly
symmetrical shape of the refractive index enhancement at Curtis
crater suggests that it may be associated with an impact ejecta pattern. We suggest that Curtis crater may be located in a stratigraphic layer of Mare Crisium which overlies a layer of material
with a higher refractive index that was exhumed by the impact
that produced Curtis. We have found a number of similar examples
of elevated refractive index centred upon the location of small
impact craters within other mare regions of the Moon. We believe
these are examples of the ‘polarisation excess’ referred to by
Shkuratov.
(b)
1.75
Peirce
1.7
1.65
1.6
Proclus
Picard
1.55
1.5
1.45
1.4
Fig. 10. A refractive index map for Mare Crisium. (a) Mare Crisium and surrounding highlands. (b) Western Mare Crisium including craters Proclus, Picard and Peirce. (b) local
terrain relief is shown and has been taken into account when calculating equigonal brightness for an Umov plot, and thence refractive index from Polarisation Anomaly. The
colour side bar shows the refractive index linear scale. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this
article.)
A. Fearnside et al. / Icarus 268 (2016) 156–171
Fig. 10(b) shows an enhanced version of the refractive index
map of Fig. 10(a) projected onto a surface relief map of a swath
of terrain. By taking account of local surface slope (illumination
angle) variations, using lunar surface topographic data made publicly available from NASA’s Lunar Orbiter Laser Altimeter (LOLA), a
more accurate equigonal brightness of the swath of terrain from
Proclus crater to (and beyond) Peirce and Picard craters was calculated. This was used in generating the Umov plot (Fig. 7(f)). The
Polarisation Anomaly and resulting refractive indices across the
swath are consequently more accurate. Over-bright Sun-facing
slopes cause some underestimation of refractive index, if not corrected in this way. The converse is true of slopes facing away from
the Sun. Note that the dark blue crescent upon the northern inner
wall of Proclus crater (Fig. 10(b)) is a consequence of CCD camera
saturation due to very high surface brightness of that Sun-facing
slope, and should be ignored. One can see immediately from
Fig. 10(b), that the refractive index for the floor regions of Picard
crater (Lat. = 14.6°, Long. = 54.7°) and of Peirce crater (Lat. = 18.3°,
Long. = 53.4°) are each markedly low against the higher refractive
index of the surrounding mare surface. This indicates that the
floors of these craters each contain highland-type materials of relatively low refractive index. Spectral studies (Head et al., 1978) and
X-ray data indicate that the floors of these craters are composed of
a material of elevated Al/Si ratio which underlies Mare Crisium,
implying that both of these craters have exhumed underlying mantle material. Elevated Al levels in minerals and glasses are generally
accompanied by a reduced refractive index value. Interestingly,
this map clearly shows a north/south asymmetry in the composition of the ejecta material of Proclus crater. Specifically, the northern portion of the ejecta pattern is rather high in refractive index,
somewhat like that of Mare Crisium, whereas the southern portion
of the ejecta is somewhat lower and comparable to that of Palus
Somni. This asymmetry is faintly indicated (though not acknowledged) in the refractive index map derived by Henry et al. (1976)
in the ultra-violet optical range. There, it can be seen that only
the northern ejecta pattern of Proclus crater displays a refractive
index comparable to that of Mare Crisium. More recently, ChangE2 orbital data has been used to generate maps of potassium abundance in and around Mare Crisium (Zhu et al., 2013) which clearly
show elevated abundances in isolation both immediately to the
north of Proclus (coincident with its northern rays) at levels comparable to mare abundances, and immediately surrounding Picard
crater. As has been suggested by others, Proclus crater appears to
have exposed an anomalous soil type (Bielefeld et al., 1978).
5.5. Factors influencing regolith refractive index
The value of refractive index of lunar minerals is dependent
upon the variation of their composition (Carli et al., 2014), as discussed by Lucey (1998) for mafic minerals and by Warell and
Davidsson (2010) for plagioclases. This is summarised in Table 7.
Furthermore, the refractive indices of lunar impact glasses,
which comprise a significant component of the regolith, have been
found to vary in value from n2 = 1.58 to n2 = 1.75 (Chao et al., 1970,
1972) in direct relation to the proportion of FeO and TiO2 present
within the glass.
Table 7
Mg# is the magnesium number defined as the molar ratio Mg# = Mg/(Mg + Fe), and
An# is the anorthite number defined as the molar ratio An# = Ca/(Ca + Na).
Mineral
Refractive index
Orthopyroxene
Clinopyroxene
Olivine
Plagioclase
n2 = 1.768 (0.118 ⁄ (Mg#))
n2 = 1.726 (0.082 ⁄ (Mg#))
n2 = 1. 827 (0.192 ⁄ (Mg#))
n2 = 1.523 + 0.0227(An#) + 0.0264(An#)2
169
5.5.1. Stratigraphy
We see in our refractive index maps that some small craters are
associated with patches of elevated refractive index. Examples of
this include Curtis crater in Mare Crisium and Reiner K in Oceanus
Procellarum. These craters may have exhumed material from a
buried basalt layer of higher FeO content than the surface mare
basalts. If so, the ejecta blanket surrounding these craters would
have a higher FeO content than the local mare surface. Given that
the magnesium number (Mg# = Mg/(Mg + Fe)) of this excavated
basalt material would be lower than that of the local surface
basalts then one might expect the refractive index of any pyroxene
and/or olivine grains and impact-induced glass within the ejecta
blanket to be elevated.
5.5.2. Impact-induced glasses
Impact processes may also contribute to the observed enhancements of refractive index due to impact-induced glass components
of the regolith. It has been observed that around 50% of the material in returned lunar soil samples consists of glasses (Ryder, 1981).
Diverse melts tend to be homogenised and the resulting glasses
represent homogenised mixtures of the original target lithologies
(Masaytis et al., 1975; Grieve, 1981). Thus, whole rock impact
melts can be expected to be simple mixtures of pre-existing target
lithologies in terms of composition. Model calculations have
shown that extensive melting occurs in a typical lunar impact
(O’Keefe and Ahrens, 1975), and melt masses of about 100 times
the impactor mass may be produced. However, studies have shown
(Schaal and Horz, 1977) that different lithologies respond to high
impact shock pressures in different ways. A prominent feature of
shock pressures is the formation of shock-metamorphic glasses
(also known as diaplectic glass, or thetomorphic glass) produced
by in situ conversion of crystal into amorphous glass of the same
composition. No melting is involved. Shock-metamorphic glasses
are only known to occur in quartz and feldspars, such as plagioclase. They possess densities and refractive indices intermediate
between those of the crystalline form of the target rock and the
impact glasses produced by melting the target rock.
5.5.3. Grain composition
Studies of lunar soil have shown that the composition of the
regolith changes with grain size. An important finding is that,
when compared to coarser grains, the smallest fraction of grains
within mare soils are consistently richer in plagioclase, and are
depleted in elements (e.g. Fe, Mg) associated with ferromagnesian
minerals (Evenson et al., 1974; Korotev, 1976; Laul et al., 1987;
Papike et al., 1981; Taylor et al., 2003). A hypothesis that may
account for this is that a greater proportion of plagioclase from disaggregated mare basalt is concentrated in the finer grain size fraction (Korotev, 1976; Papike et al., 1982; Laul et al., 1987).
Laboratory studies of impacts into basalt rocks have shown that
a preferential enrichment of plagioclase occurs in the finest grain
fraction of the disaggregated basalt particles produced by impact
shock. Plagioclase preferentially enriches the smallest grain fraction because this mineral tends to shatter into a finer grain size
than do other minerals common in basalt rocks, such as pyroxene
in particular (Horz et al., 1984). An examination of the compositions of the impact-induced grain size fractions of lunar soil samples has revealed that the abundance of plagioclase remains
relatively constant or slightly increasing with decreasing grain size.
However, the abundance of pyroxenes and olivine decreases with
decreasing grain size. Thus, a relative enrichment of finestgrained plagioclase occurs as against a relative decrease of
fine-grained pyroxenes (Taylor et al., 2003). This differential
comminution of plagioclase during regolith formation would result
in a gradual evolution of the bulk refractive index of the regolith.
When it is an immature soil, such as fresh crater ejecta, the regolith
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A. Fearnside et al. / Icarus 268 (2016) 156–171
should possess a larger average grain size and relatively little plagioclase grain enrichment. As time passes those large grains are
slowly disaggregated by impact processes into ever finer grains
which increasingly enrich the plagioclase component of the ejecta
at the expense of the pyroxene grain content. The net effect, as
young crater ejecta matures, would be to reduce its bulk refractive
index from an initially relatively higher value approximating that
associated with pyroxenes, to a lower value closer to that associated with plagioclase. The most optically relevant portion of the
regolith, for the purposes of remote sensing, is the finest grain fraction in the uppermost surface of the regolith which tends to coat
the larger grains and presents the greater optical cross section.
Thus, the evolution of the composition of this finest grain fraction will determine the evolution of the bulk optical refractive
index observed remotely. The mechanism of differential comminution would result in a local relative refractive index increase in the
fresh, large-grained and immature soils of young impact craters,
albeit temporarily. This would also be consistent with the prior
interpretations of the ‘Polarimetric Anomaly Parameter’
(Shkuratov, 1981; Dollfus, 1998; Shkuratov and Opanasenko,
1992; Geake and Dollfus, 1986; Novikov et al., 1982) and the ‘polarisation excess’ being found in fresh bright craters and caused
by anomalously large median grain sizes. We suggest that the ‘polarisation excess’ observed in those cases was not a consequence of
median grain sizes being anomalously large. Instead, it may have
been due to the fact that the anomalously large grains in question
possess a greater fraction of pyroxene which has a relatively higher
refractive index, and a lesser fraction of fine-grained plagioclase
with a relatively lower refractive index. The effect would be an
anomalously higher bulk refractive index.
5.5.4. Summary
One may conclude that a key factor controlling the value of the
bulk refractive index of the uppermost surface of the lunar regolith
is the relative contribution of: crystalline pyroxene grains; crystalline plagioclase grains; plagioclase shock-metamorphic glass;
plagioclase quenched glass; and, quenched whole-rock melt glass.
Immature regoliths are expected to be relatively abundant in the
larger-grained first factor, producing an enhancement in refractive
index. Increasing maturity enriches the regolith with each of the
other smaller-grained factors, which each reduce the refractive
index. Thus, we expect refractive index enrichment in the ejecta
blankets of young mare craters to fade over time due to this impact
mechanism which should also reduce median grain size.
6. Conclusion
The linear correlation between a peak value in the polarisation
of moonlight reflected from a lunar surface region and its associated albedo has been reproduced by each of two different mathematical models which represent the salient features of lunar
regolith grains. Both models predict that variations in grain properties such as grain diameter, bulk opacity and extent of surface
space weathering each serve to generate the linearity of the correlation enshrined by Umov’s Law. However, both models also predict that variation in the value of the real component of the
refractive index of grains has the effect of inducing a transverse
shift in the linear correlation. We refer to this as Polarisation Anomaly. This means that Polarisation Anomaly varies on the basis of
variations in grain refractive index, but is independent of the
effects of variation in grain size, opacity and space weathering. This
could provide a powerful new tool for investigating the chemistry
of the lunar regolith which mitigates the obscuring effects of space
weathering (Noble et al., 2007; Taylor et al., 2001a,b) and grain size
variations.
Polarimetric studies of different selected terrestrial materials
and lunar simulant JSC-1a have revealed good agreement with
the theoretical predictions of Polarisation Anomaly. Reinforced by
the agreement between theoretical predictions and experimental
observations, we have applied this new interpretation of Polarisation Anomaly to new telescopic observations of selected sites and
regions on the Moon in an attempt to determine the location and
extent of variations in the bulk refractive index of the grains of
the lunar regolith there. The selected sites were chosen in view
of their known mineralogical variety that, in turn, can be expected
to display a corresponding variety in refractive index. We have
indeed found strong evidence that this is the case and that Polarisation Anomaly provides a means for measuring the variation in
the refractive index of regolith grains. We propose that the existing
interpretation of the ‘Polarimetric Anomaly Parameter’, as being
due to the presence of anomalously large regolith grains, may
require a subtle revision. This revision is that Polarisation Anomaly
is not caused by an anomalous size of regolith grains per se, but is
instead caused by those grains having an anomalous mineralogy
associated with a higher (or in some cases, lower) refractive index
relative to their surroundings. Minerals of higher refractive index
may break into fragments that tend to be larger than those formed
by the breakage of minerals of lower refractive index. This mechanism may explain why the ‘Polarimetric Anomaly Parameter’ has
previously been associated with grain size variation at fresh lunar
craters.
Acknowledgments
We wish to acknowledge the kind support of Dr. K. Joy of the
University of Manchester in providing a sample of lunar simulant
JSC-1a, and of Dr. N. Bowles of the University of Oxford for measuring the refractive index of grains of this material. Our thanks go to
Chris Dudman for his help with the preparation of samples used in
the experiments described above. The JSC-1a simulant was supplied by Orbitec in support of the Chandrayaan-1 X-ray Spectrometer lunar mission activities. We also thank Kevin Kilburn FRAS and
the Manchester Astronomical Society for their steadfast encouragement and their kind permission to use the Godlee Observatory
in support of this work.
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