# Discrepancy in Grain Size Estimation of H2O Ice in the Outer Solar System
and the Interstellar Medium

A. Emran Space and Planetary Sciences, University of Arkansas, Fayetteville,
AR 72701, USA  V. F. Chevrier Space and Planetary Sciences, University of
Arkansas, Fayetteville, AR 72701, USA

###### Abstract

Widespread detection of amorphous and crystalline water (H2O) ice in the outer
solar system bodies and the interstellar medium has been confirmed over the
past decades. Radiative transfer models (RTMs) are used to estimate the grain
sizes of H2O ice from near-infrared (NIR) wavelengths. Wide discrepancies in
the estimation of H2O ice grain size on the Saturnian moons (Hansen, 2009), as
well as nitrogen (N2) and methane (CH4) ices on Kuiper belt objects have been
reported owing to different scattering models used (Emran & Chevrier, 2022).
We assess the discrepancy in the grain size estimation of H2O ice at a
temperature of 15, 40, 60, and 80 K (amorphous) and 20, 40, 60, and 80 K
(crystalline) – relevant to the outer solar system and beyond. We compare the
single scattering albedos of H2O ice phases using the Mie theory (Mie, 1908)
and Hapke approximation models (Hapke, 1993) from the optical constant at NIR
wavelengths (1 – 5 µm). This study reveals that the Hapke approximation models
– Hapke slab and internal scattering model (ISM) – predict grain size of the
crystalline phase, overall, much better compared to the amorphous phase at
temperatures of 15 – 80 K. However, the Hapke slab model estimates much
approximate grain sizes, in general, to that of the Mie model’s prediction
while ISM exhibits a higher uncertainty. We recommend using the Mie model for
unknown spectra of outer solar system bodies and beyond in estimating H2O ice
grain sizes. While choosing the approximation model for employing RTMs, we
recommend using a Hapke slab approximation model over the internal scattering
model.

Classical Kuiper belt objects (250) — Trans-Neptunian objects (1705) —
Plutoids (1268) — Radiative transfer (1335) — Surface ices (2117)

††software: NumPy (Harris et al., 2020), SciPy (Virtanen et al., 2020),
MatplotLib (Hunter, 2007), MiePython https://pypi.org/project/miepython/

##  1 Introduction

Detection of water (H2O) ice on outer solar system bodies (giant planets’
rings and their icy moons and dwarf planets), cometary bodies (originates from
the Kuiper belt and Oort cloud), and interstellar medium are well established
(see Kofman et al. 2019 for an exhaustive list of references). Solid H2O ice
in these environments may appear as either crystalline or amorphous or both
phases based on temperature, radiation histories, and their formation
temperature and pressure conditions (Mastrapa et al., 2008). Though amorphous
water ice (alternatively microscopic solid H2O ice) is considered the most
abundant form of H2O in the universe (Baragiola, 2003), the crystalline phase
has also been spotted in the many outer solar system bodies.

Phases of amorphous and crystalline H2O ice were investigated on the Galilean
moons of Jupiter using the data from NIMS instrument onboard Galileo
spacecraft (Hansen & McCord, 2004). Neptune’s moon Triton, a trans-Neptunian
object (TNO), is also assumed to host amorphous and crystalline water ice
(Cruikshank et al., 2000). Although the presence of crystalline H2O ice has
undoubtedly been reported, in a small fraction, on Pluto at the Kuiper belt
(Cook et al., 2019), its largest satellite Charon hosts both amorphous and
crystalline H2O phases (Dalle Ore et al., 2018). Molecules of amorphous H2O
ice have been detected on the icy dust grains in dense interstellar clouds
(e.g., Herbst, 2001). Due to the formation of cometary icy grains in very low
temperatures, the water ice within cometesimals is thought to be in an
amorphous phase (Raponi et al., 2016). However, the phase transition of H2O
ice from amorphous to crystalline on the cometary surface happens when comets
pass a closer distance to the Sun which leads to an activation temperature of
120 K for phase transition (e.g., Raponi et al., 2016).

Observations from different spectral wavelengths are used to study the outer
reach of the solar system. Among the wavelengths, the near-infrared (NIR) has
the most diagnostic bands and therefore have extensively been used for
characterizing ices and volatiles in outer solar system bodies (Barucci &
Merlin, 2020). While the observations of ices in the interstellar medium are
mostly accomplished from the spectral absorption signatures of infrared lights
from reddened stars (Kofman et al., 2019), the H2O ice phases in the
interstellar medium can remotely be determined by observing the 3 µm band
(Mastrapa et al., 2009). The interaction between infrared photons and H2O ices
and their resultant absorption bands and positions are dependent on its
crystalline vs amorphous phases and the temperature of the ice (Mastrapa et
al., 2009). Thus, the shapes and position of the infrared absorption bands and
their variations in response to temperature are considered as the diagnostic
for astronomical ices (e.g., Schmitt et al., 1998). At NIR wavelengths,
amorphous water ice exhibits different spectral characteristics above and
below the deposition temperature at 70 K such that the amorphous ice has
indistinguishable shapes in spectral bands below that temperature (Mastrapa et
al., 2008).

Radiative transfer models (RTMs) have been used for characterizing the
composition of the outer solar system bodies such as Kuiper belt objects
(KBOs) and TNOs (e.g., Grundy & Fink, 1991; Dumas et al., 2007; Merlin et al.,
2010; Tegler et al., 2010) and molecular clouds and accretion disk (e.g.,
Pollack et al., 1994). Wide discrepancies in the estimation of grain size have
been reported for H2O ice on Saturnian moons such as Enceladus (Hansen, 2009)
and nitrogen (N2) and methane (CH4) ices on TNOs and KBOs using different
scattering models (Emran & Chevrier, 2022). This uncertainty in grain sizes of
ices is due to the use of different single scattering albedo (w) calculations
rather than a choice of using bidirectional scattering models (Hansen, 2009).
Accordingly, Hansen (2009) compares the single scattering albedos using the
optical constant data of crystalline H2O ice at 110 K – relevant to Saturnian
moons and other icy bodies. However, temperatures at the far reach of the
solar system at the Kuiper belt, Oort cloud, and the interstellar medium are
much lower. For instance, Pluto has a mean temperature of $\mathrm{\sim}$ 40 K
(e.g., Earle et al., 2017) while the temperatures of molecular clouds in the
interstellar environment of our Milky Way galaxy are assumed to be 10 - 20 K
(e.g., Ferriere, 2001).

Moreover, while employing RTMs in the outer solar system bodies, there has
been extensive use of optical constants of crystalline and/or amorphous phases
of H2O ice and a variety of scattering models. For instance, Protopapa et al.
(2017) used the optical constant of crystalline H2O ice and employed the
equivalent slab model of Hapke (1993) for mapping the spatial distribution and
grain size of water ice (along with other ices) on Pluto. For the same
planetary body, Cook et al. (2019) used the optical constant of both amorphous
and crystalline ices while their implemented scattering model was following
the formulations of Roush (1994) and Cruikshank et al. (1998) with some
modifications. Thus, the uncertainty of the grain size estimation of amorphous
and crystalline water ice at temperatures relevant to the outer solar system
and the interstellar medium (i.e., 15 K – 80 K) using different scattering
models is warranted.

We assess the uncertainty in predicting the H2O ice grain sizes at a
temperature of 15, 40, 60, and 80 K (amorphous) and 20, 40, 60, and 80 K
(crystalline) relevant to the outer solar system bodies including the Kuiper
belt, Oort cloud, and the interstellar medium. Single scattering albedos as
being the primary cause of the varied results in the grain size estimation of
ices (Hansen, 2009), we compare the relative grain sizes of H2O ice predicted
by different single scattering models. The theory of Mie (1908) and
approximation models of Hapke (1993) are commonly used methods for calculating
the w of a material from its optical constants. Accordingly, we calculate the
w of water (amorphous and crystalline) ices by implementing the Mie theory and
widely used Hapke approximation models from the optical constants over the
wavelengths between 1 - 5 µm.

##  2 Methods

###  2.1 Optical constants

We use the NIR optical constants of H2O ice at a temperature of 15 – 80 K
(amorphous phase) and 20 – 80 K (crystalline phase). The optical constants of
amorphous and crystalline ice at NIR bands between 1.1 – 2.6 µm were collected
from Mastrapa et al. (2008) and NIR to mid-infrared bands between 2.5 – 22 µm
from Mastrapa et al. (2009). A compiled optical constant data for both ice
phases over the entire NIR wavelengths (1 – 5 µm) can be found in Mastrapa et
al. (2009). Note that the crystalline phase used in this study is assumed to
be the cubic ice form – a meta-stable H2O ice phase (Mastrapa et al., 2009).
The cubic and hexagonal crystalline ices exhibit nearly identical spectra at
infrared bands (Bertie & Whalley, 1967), and in this study, we do not
distinguish our discussion on cubic vs hexagonal ice form separately rather we
treat the crystalline phases as a unified entity. We use the cubic ice as the
representative of the crystalline ice, and hereafter, we denote the
crystalline ice as Ic and amorphous ice as Ia.

We use the optical constant data from Mastrapa et al. (2008, 2009) to maintain
consistency as prepared by the same instrumental setup and author(s), though
there have been other sources for optical constants of water ice elsewhere
too. For instance, Grundy & Schmitt (1998) prepared the optical constants for
monocrystalline (hexagonal) H2O ice samples between the wavelengths of 0.97 –
2.73 µm at temperatures of 20 to 270 K. On the other hand, Hudgins et al.
(1993) prepared the optical constants of Ia between the wavelengths of 2.5 –
20 µm at temperatures of 10 – 120 K.

###  2.2  Single scattering albedo

Photon interaction with a material grain happens at the atomic-molecular level
through the process of selective absorption, emission, and scattering (e.g.,
Shepard, 2017). The sum of photons scattered from or absorbed by a particle to
the number of photons, at a given wavelength, by its physical cross-section is
called extinction (or attenuation) efficiency, Q${}_{ext\ }$= Qsc \+
Qab${}_{.\ }$The w is the ratio of scattering efficiency and extinction
efficiency, w = Q${}_{sc\ }$/ Qext.

For planetary regolith, w is a representative of characteristic surface
properties including grain size, its optical properties, internal structure,
and partly shapes (Hapke, 1981). Consequently, the grain size estimation of
the regolith primarily depends on the characteristic value of w by the surface
particles (Hansen, 2009). Material’s w is a function of optical properties
(indices of refraction) of its single grain (e.g., Hapke, 1993; Mustard &
Glotch, 2019). Thus, w can be calculated if optical constants – real (n) and
imaginary (k) parts – of regolith medium are known. Larger particles with
moderate to larger k values result in higher absorption of incident light and
therefore, are associated with a lower w value (e.g., Shepard & Helfenstein,
2007).

Mie scattering theory can accurately estimate the w for a particle with a
spherical shape (Mie, 1908; Wiscombe, 1980). On the other hand, Hapke (1993)
formulates a variety of approximation models for the estimation of w from
material optical constants. We employed two of these approximation models –
the “espat” slab (hereafter the Hapke slab or simply slab model; Hapke 1981,
1993) and a version of the internal scattering model (ISM; Hapke 1993) – which
have widely been used in the RTMs of the outer solar system bodies. Note that
the version of the scattering model by Roush (1994) uses, indeed, a
formulation of the ISM in the original paper by Hapke (1981) and devised the
versions of surface scattering functions (Hansen, 2009).

###  2.3 Mie calculation

The scattering properties of a particle with the simplest three-dimensional
geometrics (spheres) can accurately be calculated using the Mie theory (Mie,
1908) by employing Maxwell’s equations. The formulation of Mie theory accounts
for optical constants (n \+ ik) of the particle and the fraction of particle’s
size to wavelengths under investigation (Hapke, 1993). However, the treatment
of diffraction effects in Mie’s w is required for a scattering model with
highly asymmetric phase functions (Hansen, 2009). Thus, in this study we use
$\delta$-Eddington corrected (Joseph et al., 1976) Mie single scattering
albedo, w’, calculated as (Wiscombe & Warren, 1980):

$w^{\prime}=\frac{\left(1-{\xi}^{2}\right)\ w}{1-{\xi}^{2\ }w}$ (1)

where $\xi$ is the asymmetry factor calculated by Mie theory. Following the
approach of Emran & Chevrier (2022), the Mie w was calculated by utilizing the
method of Wiscombe (1979) using the miepython routine (an open-source Python
module). The $\delta$-Eddington corrected Mie w’ was then adjusted from Mie’s
w following Eq. 1.

###  2.4 Hapke approximation models

Using the “espat” function or the slab model of Hapke (1981, 1993), the
approximate w of an equant particle can be simplified as:

$w=\frac{1}{\alpha D_{e}+1}$ (2)

where $\alpha$ is the absorption coefficient and De is the “effective particle
size”. The absorption coefficient is given by:

$\alpha=\frac{4\pi k}{\lambda}$ (3)

where k is the imaginary part of the optical constant and $\lambda$ is the
wavelength under consideration. The Hapke slab model should be applied for
materials with k $<<$1 and De can be approximated as (Hapke, 1993):

$D_{e}=\mathrm{\textrm{\^{g}}}\frac{1-S_{e}}{1-S_{i}}\textit{D}$ (4)

where D is the particle diameter, ĝ is a constant (scaling parameter of
diameter) which is roughly $=$ 1, and Se and Si are the average Fresnel
reflection coefficient for externally and internally incident light,
respectively. The derivations of the approximate equations for the calculation
of Se and Si are given in Hapke (1993) where k is set to be zero, therefore is
not considered, for the slab model. These approximate equations (Se and Si)
can also be found elsewhere in the literature that uses the Hapke models
including the study of Emran & Chevrier (2022). The calculation of w using the
ISM can be approximated as (Hapke, 1993):

$w=S_{e}+\left(1-S_{e}\right)\frac{\left(1-S_{i}\right)\textrm{$\Theta$}}{1-S_{i}\textrm{$\Theta$}}\
$ (5)

where particles’ internal-transmission function, $\Theta$ is given by (Hapke,
1993):

$\textrm{$\Theta$}=\frac{r+\mathrm{exp}\mathrm{}(-\sqrt{}(\alpha\left(\alpha+s\right))\langle
D\rangle}{1-r\
\mathrm{exp}\mathrm{}(-\sqrt{}\left(\alpha\left(\alpha+s\right)\right)\langle
D\rangle}$ (6)

where s is the near-surface internal scattering coefficient. The mean free
path of photon $\langle$D$\rangle$ as a function of n for perfectly spherical
particle can be written as (Hapke, 1993):

$\langle D\rangle\
=\frac{2}{3}\left(n^{2}-\frac{1}{n}{\left(n^{2}-1\right)}^{\frac{3}{2}}\right)D$
(7)

The internal hemispherical (diffused) reflectance, r can be given as:

$r=\frac{1-\ \sqrt{}\alpha\left(\alpha+s\right)}{1+\
\sqrt{}\alpha\left(\alpha+s\right)}$ (8)

If s is set to zero (Lucey, 1998), then r = 0 and $\mathit{\Theta}$ = e${}^{-\
\alpha}$ ⟨D⟩ (e.g., Lawrence & Lucey, 2007; Hansen, 2009; Li & Li, 2011). The
list of notations used in this study are given in Appendix (B).

##  3 Results

###  3.1 Calculated single scattering albedo

We compare the w spectra of 10 µm radii Ia and Ic particles to assess the
difference in calculated albedo spectra at that specific diameter calculated
using the Mie and Hapke approximation models over the NIR bands (Fig. 1).
Following the approach of Emran & Chevrier (2022), we apply the Savitzky-Golay
filter (Savitzky & Golay, 1964) to smoothen the small spikes in calculated w
curves over the wavelengths. We implement the filter algorithm with a third
order polynomial fit. Note that all plots used in this study were smoothened
using the Savitzky-Golay filter, if not mentioned otherwise.

Figure 1: Calculated single scattering albedo plots for 10 µm radii particles
of amorphous (left column) and crystalline (right column) H2O ice at the NIR
wavelengths. The subplots in the rows are the measurements at different
temperatures where each dashed solid line is $\delta$-Eddington corrected Mie,
the dotted green is Hapke slab, and dashed yellow is ISM spectra.

The w plots from the Hapke approximation models follow the spectra generated
by the Mie model with varying degrees of closeness at different NIR
wavelengths. In the case of Ia at every investigated temperature (the left
column in Fig. 1), the Hapke slab models follow much closer, overall, to the
Mie model spectra compared to the ISM model spectra, except around the broader
3.0 µm region. At shorter wavelengths up to 2.5 µm, the slab model produces
slightly higher w values than the Mie model, notably around weak overtone 1.5
and 2.0 µm bands at every investigated temperature. This represents a
prediction of slightly smaller H2O grains by the slab model than the Mie model
(Hansen, 2009) over the shorter wavelengths at weak overtone 1.5 and 2.0 µm
bands. Neither of the approximation model spectra does mimic the Mie spectra
at 3 µm wavelengths, except the ISM spectra do partially follow the shape of
the Mie spectra on the bands past 3 µm. The inability of the slab model in
predicting the 3 µm absorption band is likely due to having higher k values at
the wavelengths.

For Ia at the broader 4.5 µm absorption band, the slab model either exactly
follows the Mie spectra (at 15K) or renders slightly higher w values than the
Mie spectra (at 40, 60, and 80K). This represents that the slab model at these
wavelengths either exactly predicts the same grain sizes or produces slightly
smaller grain sizes than Mie’s prediction, respectively. In contrast, at every
temperature, the ISM renders much lower w than the Mie spectra at 4.5 µm band
– indicating larger grain size prediction by the ISM. At longer wavelengths
around 3.5 and 5 µm, both approximation models show smaller w values than the
Mie model – meaning a larger grain size prediction by the approximation models
at these bands. However, at the latter band, the grain size prediction is
larger than the former band.

In the instance of Ic at every temperature (the right column in Fig. 1), the
slab model accurately mimics the Mie model at the weak overtone 1.5 µm while
it shows slightly higher w values at the weak overtone 2.0 µm. This represents
an accurate grain size prediction at 1.5 µm and a slightly smaller grain size
prediction at 2.0 µm by the Hapke slab model. The ISM, on the other hand,
predicts much smaller grain sizes at 1.5 and 2.0 µm. The reflection peak at
3.1 µm does not represent by either of the approximation models due to having
a higher k value at this band (Hansen, 2009). At the 4.4 µm absorption band,
the slab model predicts smaller grain sizes while the ISM predicts larger
grain sizes compared to Mie’s prediction. However, both the approximation
models follow the same path and show much smaller w values at 3.5 and 5 µm
bands – indicating a much larger grain size prediction by the approximation
models. Compared to Ia, the spectra of Hapke approximation models in Ic show a
much larger difference than Mie’s value at 3.5 µm.

###  3.2 Relative grain size estimation

We assess the uncertainty in the predictions of grain sizes by the Hapke
approximation models for the ice phases at the investigated temperatures. A
finely spaced range of grain sizes at radii of 1, 10, 100, and 1000 µm was
used to show the relative discrepancies of grain size prediction at NIR bands.
We choose this finely spaced range because it covers the grain sizes, from
microns (µm) to millimeters (mm), found in the outer solar system bodies. For
instance, fine-grain H2O ice with a diameter of $\mathrm{<}$ 1 µm has been
predicted on Pluto (Cook et al., 2019). The presence of much larger ices grain
sizes is also believed to be present in outer solar system bodies such as icy
moons and rings of Saturn (e.g., Filacchione et al., 2012) and Pluto (e.g.,
Cook et al., 2019).

Accordingly, we first compute the $\delta$-Eddington corrected Mie w’ at these
fixed grain radii over the NIR bands. Once the Mie w’ is  calculated, we
estimate the corresponding grain sizes from each of the Hapke approximation
models by fitting the Mie spectra over the wavelengths. To this end, we solve
the inverse of ISM and Hapke slab model for D to fit the corresponding Mie w’
values. We use Powell’s hybrid method (Powell, 1970; Chen & Stadtherr, 1981)
to solve the non-linear function of Hapke slab and ISM (Eq. 2 and 5) in
finding the solution of D. The relative grain size prediction was then
determined by normalizing the predicted grain sizes by the approximation
models to the grain sizes of Mie model.

We estimate the statistics, medians and their 16% quantile as a lower
1$\sigma$ error bar and 84% quantile as an upper 1$\sigma$ error bar, of
relative discrepancies in predicted grain sizes using the approximation models
at different particle radii for Ia (Table 1a) and Ic (Table 1b) at the
investigated temperatures. As additional supporting information, we also list
the mean $\mathrm{\pm}$ 1$\sigma$ standard deviation relative discrepancies in
predicted grain sizes in the Appendix (Table 2). The statistics were derived
from the normalized values of the estimated grain sizes predicted by the slab
model and ISM to the Mie model. A value of 1.0 represents exact prediction
while the higher or lower values imply an over or underestimation,
respectively, of the grain sizes by the respective approximation models. The
plots for predicted relative gain sizes using the Hapke slab and ISM for Ia
and Ic are given in Fig. 2 and Fig. 3, respectively.

Table 1: The statistics - medians and their 16% quantile as a lower 1$\sigma$
error bar and 84% quantile as an upper 1$\sigma$ error bar - of relative
discrepancies in predicted grain sizes using the Hapke slab and ISM calculated
at particle radii of 1, 10, 100, and 1000 µm for amorphous (a) and crystalline
(b) H2O ice at different temperatures.  a.
---
Grain radii (µm) | Amorphous H2O ice∗
15 K | 40 K | 60 K | 80 K
Slab | ISM | Slab | ISM | Slab | ISM | Slab | ISM
1 | ${\mathrm{1.07}}^{+1.04}_{-0.25}$ | ${\mathrm{1.28}}^{+1.08}_{-0.29}$ | ${\mathrm{1.07}}^{+0.97}_{-0.25}$ | ${\mathrm{1.28}}^{+1.02}_{-0.29}$ | ${\mathrm{1.07}}^{+0.96}_{-0.25}$ | ${\mathrm{1.29}}^{+1.04}_{-0.30}$ | ${\mathrm{1.08}}^{+0.92}_{-0.26}$ | ${\mathrm{1.29}}^{+1.03}_{-0.30}$
10 | ${\mathrm{1.19}}^{+0.06}_{-0.34}$ | ${\mathrm{1.44}}^{+0.05}_{-0.56}$ | ${\mathrm{1.20}}^{+0.07}_{-0.29}$ | ${\mathrm{1.45}}^{+0.05}_{-0.57}$ | ${\mathrm{1.21}}^{+0.06}_{-0.31}$ | ${\mathrm{1.46}}^{+0.05}_{-0.58}$ | ${\mathrm{1.21}}^{+0.06}_{-0.26}$ | ${\mathrm{1.45}}^{+0.05}_{-0.58}$
100 | ${\mathrm{1.46}}^{+0.58}_{-0.12}$ | ${\mathrm{1.66}}^{+0.35}_{-0.10}$ | ${\mathrm{1.46}}^{+0.58}_{-0.26}$ | ${\mathrm{1.66}}^{+0.48}_{-0.09}$ | ${\mathrm{1.47}}^{+0.62}_{-0.26}$ | ${\mathrm{1.67}}^{+0.45}_{-0.09}$ | ${\mathrm{1.47}}^{+0.60}_{-0.35}$ | ${\mathrm{1.67}}^{+0.51}_{-0.09}$
1000 | ${\mathrm{1.54}}^{+1.08}_{-1.37}$ | ${\mathrm{1.67}}^{+4.34}_{-1.08}$ | ${\mathrm{1.54}}^{+1.09}_{-1.38}$ | ${\mathrm{1.67}}^{+4.34}_{-1.19}$ | ${\mathrm{1.55}}^{+1.11}_{-1.37}$ | ${\mathrm{1.68}}^{+4.35}_{-1.23}$ | ${\mathrm{1.55}}^{+1.10}_{-1.40}$ | ${\mathrm{1.68}}^{+3.87}_{-1.26}$
b.
Grain radii (µm) | Crystalline H2O ice∗
20 K | 40 K | 60 K | 80 K
Slab | ISM | Slab | ISM | Slab | ISM | Slab | ISM
1 | ${\mathrm{0.88}}^{+0.67}_{-0.19}$ | ${\mathrm{1.03}}^{+0.76}_{-0.22}$ | ${\mathrm{0.87}}^{+0.67}_{-0.19}$ | ${\mathrm{1.03}}^{+0.76}_{-0.22}$ | ${\mathrm{0.88}}^{+0.67}_{-0.19}$ | ${\mathrm{1.03}}^{+0.75}_{-0.22}$ | ${\mathrm{0.87}}^{+0.67}_{-0.19}$ | ${\mathrm{1.03}}^{+0.75}_{-0.22}$
10 | ${\mathrm{1.10}}^{+0.08}_{-0.07}$ | ${\mathrm{1.30}}^{+0.07}_{-0.44}$ | ${\mathrm{1.09}}^{+0.07}_{-0.06}$ | ${\mathrm{1.30}}^{+0.07}_{-0.44}$ | ${\mathrm{1.09}}^{+0.07}_{-0.07}$ | ${\mathrm{1.30}}^{+0.07}_{-0.44}$ | ${\mathrm{1.09}}^{+0.07}_{-0.07}$ | ${\mathrm{1.30}}^{+0.07}_{-0.44}$
100 | ${\mathrm{1.24}}^{+0.65}_{-0.54}$ | ${\mathrm{1.46}}^{+0.40}_{-0.03}$ | ${\mathrm{1.24}}^{+0.65}_{-0.54}$ | ${\mathrm{1.46}}^{+0.40}_{-0.03}$ | ${\mathrm{1.25}}^{+0.64}_{-0.51}$ | ${\mathrm{1.46}}^{+0.41}_{-0.03}$ | ${\mathrm{1.24}}^{+0.64}_{-0.51}$ | ${\mathrm{1.46}}^{+0.41}_{-0.03}$
1000 | ${\mathrm{1.34}}^{+0.80}_{-1.27}$ | ${\mathrm{1.47}}^{+3.57}_{-1.25}$ | ${\mathrm{1.34}}^{+0.81}_{-1.27}$ | ${\mathrm{1.67}}^{+3.64}_{-1.25}$ | ${\mathrm{1.34}}^{+0.81}_{-1.26}$ | ${\mathrm{1.47}}^{+3.78}_{-1.24}$ | ${\mathrm{1.34}}^{+0.81}_{-1.26}$ | ${\mathrm{1.47}}^{+3.78}_{-1.24}$
Figure 2: Predicted relative gain sizes (normalized) using the Hapke slab
(left column) and ISM (right column) calculated at particle radii of 1, 10,
100, and 1000 µm for amorphous H2O ice. The subplots in the rows are the
measurements at different temperatures.  Figure 3: Predicted relative gain
sizes (normalized) using the Hapke slab (left column) and ISM (right column)
calculated at particle radii of 1, 10, 100, and 1000 µm for crystalline H2O
ice. The subplots in the rows are the measurements at different temperatures.

The predicted grain sizes by both approximation models are much closer,
overall, to Mie’s prediction for the Ic than the Ia. This is supported by the
better-predicted medians values (much closer to 1) in the relative grain size
estimation by the Ic compared to the Ia (Table 1) – also visually evident in
Fig. 2 and 3. For instance, between the ice phases at the corresponding grain
radii (e.g., 10 µm) and temperatures (e.g., 40 K), the median values are much
better by both approximation models for Ic compared to the Ia. An exception in
this comparison is that for 1 µm grain the slab model’s prediction is slightly
closer to 1 for Ia than Ic; however, their 16 and 84% quantile error bars are
higher. The mean values (Table 2 in Appendix) between the ice phases are also
consistent – although not completely at every temperature and grain size –
with the median values, except for smaller grains ($\mathrm{\leq}$ 10 µm) at
lower temperatures ($\mathrm{\leq}$ 40 K). Therefore, it can be concluded that
the Hapke approximation models predict much better, in general, grain sizes
for the Ic than the Ia phase.

Between the approximation models, the slab model shows better approximations
(much closer to 1) than the ISM for both ice phases – evident from the median
(except for Ic at 1 µm) and mean $\mathrm{\pm}$ 1$\sigma$ values at every
corresponding temperature and grain size. Even though the slab model-predicted
median values for 1 µm particles are slightly farthest for Ic, their 16 and
84% quantile error bars indicate a lower discrepancy than the ISM. The plots
for the relative discrepancies in grain size of Ia (Fig. 2) and Ic (Fig. 3)
also support that the slab model predicts a much better approximate result. In
contrast, the discrepancies in grain size estimation by ISM are much higher –
consistent with the result of Hansen (2009). However, around the absorption
peak of the 3.1 µm band, neither of the approximation models predicts good
results.

The 1 µm curve at every temperature and ice phase appears a continuous rise at
shorter wavelengths up to $\mathrm{\sim}$2.8 µm. This continuous rise at
shorter bands is largely due to the Rayleigh effect on w from the Mie model
that happens at wavelengths close to or smaller than the grain size (Hansen,
2009). The Hapke approximation models do not account for the Rayleigh effect
of scattering models. Thus, even if the median values of the 1 µm curve show a
good prediction (Table 1), the mean and 1$\sigma$ standard deviation in
predicated grain sizes (Table 2 in Appendix) show one of the highest
discrepancies in grains size estimation by the approximation models. The 10 µm
curve for each ice phase and temperature illustrates better prediction than
other grains size curves. The median values are also in agreement with the
statement since the medians are close to 1. This is further supported by
having the predicted mean value closest to 1 for both models compared to other
grains sizes considered.

Though the 100 µm curves show less uncertainty (relatively flat) in grain size
estimation at shorter wavelengths by the ISM than the slab model, at longer
wavelengths ($\mathrm{>}$3 µm) the scenario is opposite. Consequently, the 100
µm curves from the slab model show much better, overall, than the ISM’s
prediction for the corresponding water ice phases and temperatures. This is
further supported by the median and means $\mathrm{\pm}$ 1$\sigma$ of
predicted grain sizes for both ice phases by the approximation models at that
grain size. The 1000 µm curve from the slab model for both ice phases show a
scattered prediction of grains size among the model’s predicted grain sizes –
also evident by having the highest median values among grain sizes by the slab
model. The corresponding 1000 µm curve by ISM, in contrast, predicts much
higher discrepancies.

For larger particles, the approximation models predict much smaller grain
sizes at longer wavelengths (Fig. 2 and 3). This is because the approximation
models do not reproduce the characteristic higher saturation level of the Mie
model (Hansen, 2009). Accordingly, the difference in grain sizes for the 1000
µm curve by ISM is much higher than that of the 1 µm curve where Rayleigh
effects render higher uncertainty in grain size prediction by the
approximation model. The mean and median values of predicated grain sizes by
the ISM’s 1000 µm curve for both phases and every temperature are the largest
among all ISM’s predicted grain sizes.

###  3.3 Chracteristic absorption coefficient

We plot the $\alpha$ distribution for Ia and Ic at different temperatures as a
function of wavelengths (Fig. 4). At every temperature, both phases have a
broad and the highest $\alpha$ peak of around 3.0 µm wavelengths with a slight
shift toward shorter wavelengths by the Ia than the Ic phase. The higher
absorption peak at 3.0 µm is due to the higher k values at these bands which
makes the inability to gain size prediction by the approximation models. This
evidence supports the assumption that the slab model should be used if k
$\mathrm{<}$ $\mathrm{<}$ 1 (Hapke, 1981), and consequently a higher k value
leads to failure in predicting accurate grain size. A blue shift in $\alpha$
spectra of Ia compared to the Ic phase is also evident at the weak 2 µm band.
This characteristic $\alpha$ distribution for H2O ice phases coincides with
the fact that Ic has much stronger, sharper, and red-shifted at infrared bands
than Ia (Schmitt et al., 1998).

Higher values of $\alpha$ are seen at the longer wavelengths for both ice
phases. However, a peak of $\alpha$ happens around 4.5 µm for Ia while this
peak shifts toward shorter wavelengths at 4.4 for Ic – in agreement with
Mastrapa et al. (2009). The effects of these peaks are evident in the shapes
of relative grain size curves for larger particles i.e., 100 and 1000 µm
curves in Fig. 2 and 3. Notably, the Ia shows comparatively much lower
$\alpha$ values around the band of 4.5 µm region than the Ic at every
investigated temperature. This relatively higher $\alpha$ value at this
wavelength for Ic compared to Ia is also apparent in the relative grain-size
curve of the 1000 µm particle (Fig. 2 and 3) where the Ic reveals a higher
inconsistency in grain size prediction than the Ia at this wavelength for both
approximation models.

The $\alpha$ of pure CH4 and N2:CH4 (N2 saturated with CH4) systems, the other
abundant volatile ices found on KBOs and TNOs surfaces, at NIR wavelengths
(see Fig. 3 of Emran & Chevrier 2022) have a much lower $\alpha$ than water
ice phases. A simple interpretation of this fact is that may be the H2O ice
has higher absorption and lower reflectance compared to the pure CH4 and
N2:CH4 ices on the TNOs (e.g., Triton) and KBOs (e.g., Pluto, Eris, and
Sedna). The results above show that the higher discrepancies in predicated
grains sizes for larger particles are largely aligned to the wavelengths with
higher $\alpha$ values. Accordingly, the estimation of H2O ice grain size
using the Hapke approximation models for larger particles may be susceptible
to higher uncertainty than that of CH4 and N2:CH4 ices at the Kuiper belt
(Emran & Chevrier, 2022).

The higher uncertainty in H2O ice than CH4 and N2:CH4 ices is further
supported by the fact that the $\alpha$ of Ia at the 4.5 µm absorption band
(and slightly at 1.5 and 2 µm bands) increases with the increase of
temperature from 15 to 80 K (Fig. 4). This $\alpha$ vs temperature
characteristic of Ia may likely be the cause of the trend of higher
discrepancies in grain size estimation with the higher temperatures at these
bands (Fig. 2). In contrast, Ic does not show any substantial change in
$\alpha$ values at the 4.4 µm absorption band due to temperature changes,
although $\alpha$ slightly decreases with temperature increase. Therefore, in
the case of Ic, there are no substantial differences in grain size estimations
at this band between the temperatures (Fig. 3). However, we acknowledge that
this statement on uncertainty in different grain size estimations of H2O, CH4,
and N2:CH4 ices is purely based on the simplest interpretation from the
comparison of the $\alpha$ values between the ice species at NIR bands, while
other factors involved, the scenario may not be true.

Figure 4: Absorption coefficient of amorphous and crystalline H2O ice at
different temperatures over NIR wavelengths. Both phases of H2O have a broad
absorption coefficient peak of around 3.0 µm with a little shift toward
shorter wavelengths by the Ia. Another absorption peak is seen at 4.5 µm for
Ia while this peak shifts toward shorter wavelengths at 4.4 for Ic with much
higher $\alpha$ values. The higher absorption peak for Ic compared to Ia at
longer wavelengths is associated with the higher k values at these
wavelengths.

##  4 Discussion

Water ice grain size estimation for outer solar system bodies from near-
infrared wavelengths has reportedly been done using the optical constants of
Ic and/or Ia by utilizing RTMs. However, estimation of grains sizes on
planetary regolith depends primarily on the calculation of w rather than a
choice of bidirectional scattering models (Hansen, 2009). The Hapke
approximation models are a well-recognized method for deriving the w and,
therefore, the models have been used in estimating surface grain sizes of icy
bodies. We assess the relative grain size prediction by widely used Hapke
approximation models for water ices phases found in the outer solar system and
beyond. Accordingly, we use a range of fixed grains sizes (1, 10, 100, and
1000 µm) in Mie calculation and then determine the corresponding grain sizes
from the Hapke approximation models to fit the Mie spectra at different NIR
wavelengths.

We use the Hapke models for predicting grain sizes of a 1 µm radii even though
the approximation model assumes that the particle size should be much larger
than the wavelengths (Hapke, 1981). This study also uses wavelengths close to
or even larger than a grain size of 1 µm. Indeed, we test a wide range of
grain sizes where the 1 µm grain size is just an instance, while particles
with sizes much larger than the wavelengths were also considered. The
rationale for using 1 µm radii is that there has been reported the presence of
H2O ice with similar sizes on the surface of outer solar system bodies such as
KBOs (e.g., Cook et al., 2019). Moreover, the use of grain size close to 1 µm
emphasizes the Rayleigh effect that happens at grains that have sizes close to
wavelengths for the outer solar system surface ices (e.g., Hansen, 2009; Emran
& Chevrier, 2022). And by taking smaller particle sizes, we evaluate the grain
size prediction discrepancies at wavelengths close to or even larger than the
grain size of 1 µm. On top of that our result shows how deviant, as it is
expected, grain size prediction the approximation models can produce (Fig. 2
and 3) when the wavelengths are close to or even larger than the grain size.

The predicted grain sizes by the Hapke approximation models vary over the
investigated wavelengths due to the optical constants (and, therefore,
different absorption coefficient values) at different NIR bands and the
formulation of the approximation models used (see Eq. 2 and 5). In simple
words, the approximation models predict different grain sizes based on optical
constant values (n \+ ik) at different wavelengths. Accordingly, assessing the
predictions of these varying grain sizes at different wavelengths by the
approximation models was the motivation of the study and thus, we evaluate the
discrepancies of predicting varying grain sizes by the scattering models. As
mentioned above, we only consider the optical constant of cubic crystalline
ice as a generalization of crystalline ice. However, there are only subtle
differences in spectra between the cubic and hexagonal phases (Bertie &
Whalley, 1967), and thus, we emphasize that the crystalline phases are
substitutable (Mastrapa et al., 2009) and the use of Ic for generalization of
the crystalline ice is reasonable.

Between the ice phases, the grain size estimation for Ic using the Hapke
approximation models results in a much closer, in general, estimate to Mie’s
prediction. Conversely, the Hapke approximation models show higher
discrepancies in grain size estimation for Ia. The selection of temperatures
for the ice phases is also a consideration. The discrepancy in grain size
estimation of larger Ia particles increases with temperatures from 15 to 80 K
at longer wavelengths, whereas these discrepancies in grain size estimation
are negligible for Ic between the temperature range. However, grain size
estimation for larger particles at the broad 4.4 - 4.5 µm absorption band
leads to higher discrepancy for Ic compared to the Ia. The relative
discrepancies of the grain size estimation is better (close to 1) for the
Hapke slab model than ISM at cryogenic temperatures. This indicates Hapke slab
is a better approximation model for deriving the water ice grain sizes in the
icy bodies in the far reach of the solar system. Our result is consistent with
the results of previously analyzed Hansen (2009) using the optical constant of
Ic at a temperature of 110 K.

Owing to the Rayleigh effects on single scattering albedo, neither of the
approximation models predicts accurate grain size estimation results for the
particles with diameters less than or close to the wavelengths
($\mathrm{\leq}$ 5 µm). For both Ia and Ic, grain size estimations for 10 µm
radii H2O ice are best predicted by both Hapke approximation models. Though
grain size prediction by the approximation models for larger particles shows
higher discrepancies largely at the wavelengths with higher absorption
coefficient values, the uncertainty in the instance of ISM is much higher than
the Hapke slab model. The higher discrepancies of grain size estimation for
larger particles by ISM can be compromised by adjusting the free parameters
used in calculating single scattering albedo, for instance, the study of Roush
et al. (2007) used a higher value of s to fit modeled spectra.

While formulating the “espat” function for the slab model, Hapke (1981) used a
scaling factor of ĝ = 2/3 for particle diameter. However, the scaling factor
of diameter in the Hapke slab model varies between 3/4 - 4/3 which can, in
principle, result in a difference in grain size estimation by the slab model
(Hansen, 2009). Likewise, the difference in grain size estimation by ISM can
also be produced by scaling factor to the particle diameter through
calculating the mean free path of the photon
$\mathrm{\langle}$D$\mathrm{\rangle}$ as seen in Eq. 7. There have been
different approximate expressions for the relationship between D and
$\mathrm{\langle}$D$\mathrm{\rangle}$. This relationship is approximated as
$\langle$D$\rangle$ $\mathrm{\cong}$ 0.9D or 2D/3 (e.g., Hapke, 2012; Melamed,
1963) for spherical particles while it’s $\langle$D$\rangle$ = 0.2D for
irregular particles (Shkuratov & Grynko, 2005). Hence, the grain sizes
estimation by the approximations models can be improved by carefully adjusting
the scale factor for particle diameters. We use a scale factor of 1 for the
slab model while the mean free path of a photon in ISM was calculated as a
function of the diameter and real part of the refraction index.

An obvious difference between the ISM and the slab models is the calculation
of average Fresnel reflection coefficients – the former model employs data
from both n and k parts while the latter uses only n part of the refractive
indices (e.g., Emran & Chevrier, 2022). Since the slab model can be applied
when k is very small (Hapke, 1981, 1993) and therefore, at wavelengths with a
substantially higher k value can lead to a higher difference in calculated
results between the models. Note that if the approximation model assumes the
internal scattering coefficient, s = 0 (as used by many existing studies, for
example, Lucey 1998; Lawrence & Lucey 2007) then the internal diffused
reflectance, r equals to be zero as well. However, Sharkey et al. (2019)
expressed the relationships between the s and D (“effective diameter”) as s =
1/D and the number of scattering events (sD) = 1 within a single grain. Thus,
this relationship indicates that s cannot be zero (Emran & Chevrier, 2022).

As has previously been mentioned that Mie’s theory (Mie, 1908) can accurately
calculate the scattering properties of uniform (shapes) particles. Further,
besides application to scattering properties for the equant particles of Mie
spheres, the Mie scattering model has been devised to be satisfactorily
employed for particles with varied “equivalent” non-spherical shapes if each
non-spherical particle can be represented by assembles of spheres with the
same volume-to-surface-area ratio as the non-spherical particle (e.g.,
Grenfell & Warren, 1999; Grenfell et al., 2005). In other words, roughly
similar scattering properties have been reported by irregular particles to
that of the scattering properties calculated by the similar-sized spherical
particles using Mie’s theory (Hapke, 1981). This implies that the scattering
properties of Mie theory for spherical particles can be mimicked to irregular
particles (Neshyba et al., 2003), and thus application to Mie theory in the
grain size estimation of a Mie’s equant particle is somewhat analogous to non-
spherical/ irregular grains (e.g., Hapke, 1981; Hansen, 2009).

While both Hapke approximation models do not account for the Rayleigh effect
on the single scattering albedo that happens to grain size close to the
wavelengths ($\mathrm{\leq}$5 µm), the Mie theory appropriately accounts for
this shortcoming. Though the rough average difference between the predicted
grain sizes by the ISM and slab model is about $\mathrm{\sim}$ 10 – 20% (based
on the median values in Table. 1) for a wide range of particles, we emphasize
that it’s still an over/underestimation by one approximation model to another.
And our study shows which approximation model predicts much better results
than others in estimating the grains sizes of Ia and Ic found in the vast
region of the outer solar system. A better estimation of H2O ice grain sizes
by the slab model than ISM is also in agreement with the grain size estimation
of the abundant surface volatile of N2 and CH4 ices on TNOs and KBOs surfaces
(Emran & Chevrier, 2022). Thus, the Hapke slab model is better equipped for
the implementation of RTMs in the approximate grain size estimation for a
range of surface ices (H2O, N2, and CH4) in the outer solar system bodies.

We recommend using the Mie formulation of the single scattering albedo for
estimation of H2O ice (Ia and Ic) grain sizes for unknown spectra of outer
solar system bodies and beyond. When choosing the approximation models, we
recommend using the Hapke slab model over ISM since our result concludes that
the slab model better approximates the H2O grain size prediction, overall,
than the internal scattering model at 1 – 5 µm wavelengths. Note that the
single scattering albedo may not be readily available from the measurements of
spacecraft observations. However, the radiance or reflectance measurement by
spacecraft observation can be converted to single scattering albedo using the
radiative transfer model of Hapke (1993).

##  5 Conclusion

The mechanical strengths and thermal properties (e.g., temperatures) of
planetary surfaces are directly influenced by the grain size of planetary
regolith (e.g., Gundlach & Blum, 2013). Thus, an accurate grain size
estimation of the outer solar system bodies is very important for an improved
understanding of the surface volatile transport and thermophysical modeling of
the planetary bodies. The icy bodies in the outer solar system host a
widespread presence of water ice on their surfaces. Understanding the grain
sizes of planetary regolith supports landing or sample return missions
(Gundlach & Blum, 2013), and thus accurate information about the water ice
grains sizes is crucial for a future mission to the outer reach of the solar
system.

We assess the discrepancies in the grain size estimation of amorphous and
crystalline H2O ice at cryogenic temperatures from 15 – 80 K at a
$\mathrm{\sim}$20 K interval. The relative grain size prediction by the Hapke
approximation models was assessed concerning ice phases, grain sizes, and
temperatures relevant to the icy zones of the solar system and the
interstellar space. Having a satisfactory application of Mie theory to an
equant as well as “equivalent” non-spherical particles, we recommend using Mie
formulation for unknown spectra of the outer solar system bodies. However, if
approximation models are opted to use, our study favors using the slab model
as a better option than the internal scattering model.

We demonstrate which model works better for the approximate grain size
estimation of phases of water ice found in the outer solar system.
Incorporation with the results from Emran & Chevrier (2022), we conclude which
approximate model is better in predicting the grain sizes of a variety of
surface ices found in the outer solar system bodies, more specifically on
trans-Neptunian objects and Kuiper belt objects. Thus, this study put forward
a prescription for future studies in choosing scattering models in estimating
the water ice grain sizes in the outer solar system and beyond using the
radiative transfer models at near-infrared bands.

The authors would like to thank anonymous reviewers for their useful comments.

##  Appendix A Statistics

The mean and $\mathrm{\pm}$ 1$\sigma$ standard deviation of relative
discrepancies in grain sizes are predicated by the approximation models

Table 2: The statistics - mean and $\mathrm{\pm}$ 1$\sigma$ standard deviation
- of relative discrepancies in predicted grain sizes using the Hapke slab and
ISM calculated at particle radii of 1, 10, 100, and 1000 µm for amorphous (a)
and crystalline (b) H2O ice at different temperatures.  a.
---
Grain radii (µm) | Amorphous H2O ice∗
15 K | 40 K | 60 K | 80 K
Slab | ISM | Slab | ISM | Slab | ISM | Slab | ISM
1 |  1.89$\mathrm{\pm}$3.06 |  2.07$\mathrm{\pm}$2.96 |  1.96$\mathrm{\pm}$3.42 |  2.25$\mathrm{\pm}$4.13 |  2.08$\mathrm{\pm}$3.97 |  2.43$\mathrm{\pm}$5.14
|  2.11$\mathrm{\pm}$4.10 |  2.44$\mathrm{\pm}$5.22
10 |  1.21$\mathrm{\pm}$0.74 |  1.40$\mathrm{\pm}$0.92 |  1.23$\mathrm{\pm}$0.82 |  1.14$\mathrm{\pm}$0.98 |  1.26$\mathrm{\pm}$0.95 |  1.42$\mathrm{\pm}$1.06 |  1.27$\mathrm{\pm}$0.98 |  1.43$\mathrm{\pm}$1.10
100 |  1.58$\mathrm{\pm}$0.72 |  2.05$\mathrm{\pm}$1.62 |  1.57$\mathrm{\pm}$0.75 |  2.06$\mathrm{\pm}$1.59 |  1.59$\mathrm{\pm}$0.80 |  2.04$\mathrm{\pm}$1.57 |  1.59$\mathrm{\pm}$0.81 |  2.06$\mathrm{\pm}$1.59
1000 |  1.51$\mathrm{\pm}$1.00 |  2.62$\mathrm{\pm}$2.44 |  1.51$\mathrm{\pm}$1.02 |  2.57$\mathrm{\pm}$2.47 |  1.52$\mathrm{\pm}$1.03 |  2.58$\mathrm{\pm}$2.48 |  1.50$\mathrm{\pm}$0.99 |  2.41$\mathrm{\pm}$2.29
b.
Grain radii (µm) | Crystalline H2O ice∗
20 K | 40 K | 60 K | 80 K
Slab | ISM | Slab | ISM | Slab | ISM | Slab | ISM
1 |  2.07$\mathrm{\pm}$5.76 |  2.19$\mathrm{\pm}$5.99 |  2.05$\mathrm{\pm}$5.64 |  2.17$\mathrm{\pm}$5.89 |  1.97$\mathrm{\pm}$5.15 |  2.13$\mathrm{\pm}$5.62 |  1.97$\mathrm{\pm}$5.15 |  2.13$\mathrm{\pm}$5.62
10 |  1.24$\mathrm{\pm}$1.35 |  1.32$\mathrm{\pm}$1.08 |  1.24$\mathrm{\pm}$1.3 |  1.31$\mathrm{\pm}$1.05 |  1.22$\mathrm{\pm}$1.20 |  1.30$\mathrm{\pm}$0.99 |  1.22$\mathrm{\pm}$1.20 |  1.30$\mathrm{\pm}$0.99
100 |  1.44$\mathrm{\pm}$1.05 |  1.74$\mathrm{\pm}$1.25 |  1.44$\mathrm{\pm}$1.04 |  1.74$\mathrm{\pm}$1.24 |  1.43$\mathrm{\pm}$0.99 |  1.75$\mathrm{\pm}$1.26 |  1.43$\mathrm{\pm}$0.99 |  1.75$\mathrm{\pm}$1.26
1000 |  1.28$\mathrm{\pm}$0.86 |  2.30$\mathrm{\pm}$2.21 |  1.27$\mathrm{\pm}$0.85 |  2.31$\mathrm{\pm}$2.21 |  1.27$\mathrm{\pm}$0.84 |  2.31$\mathrm{\pm}$2.22 |  1.22$\mathrm{\pm}$0.84 |  2.31$\mathrm{\pm}$2.22

##  Appendix B Notation

List of notation and symbols used in this paper

$\langle$D$\rangle$ mean free path of a photon

D particle diameter

De effective particle sizes

$\xi$ asymmetry parameters of Mie theory

ĝ a constant, roughly = 1

k imaginary part of the refractive index

n real part of the refractive index

r internal diffused reflectance

s internal scattering coefficient

Se Fresnel reflection coefficient for externally incident light,

Si Fresnel reflection coefficient for internally incident light,

w single scattering albedo

w’ $\delta$-Eddington Mie single scattering albedo

$\alpha$ absorption coefficient

$\lambda$ wavelength

$\Theta$ internal transmission coefficient

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