[orcid=0000-0002-8780-8226] [1]

[orcid=0000-0003-1236-1228]
[orcid=0009-0008-6736-557X][orcid=0009-0008-0789-2052]
[orcid=0000-0003-1591-6647]

1]organization=Department of Physics, University of California San Diego,
addressline=9500 Gilman Drive, city=La Jolla, postcode=92093-0424, state=CA,
country=USA

2]organization=Department of Astronomy and Astrophysics, University of
California San Diego, addressline=9500 Gilman Drive, city=La Jolla,
postcode=92093-0424, state=CA, country=USA

3]organization=Department of Physics and Astronomy, Wellesley College,
addressline=106 Central Street, city=Wellesley, postcode=02481, state=MA,
country=USA

4]organization=Geodesy and Geophysics Lab, NASA Goddard Space Flight Center,
addressline=8800 Greenbelt Rd., city=Greenbelt, postcode=20771, state=MD,
country=USA 5]organization=Department of Physics, University of Maryland
Baltimore County, addressline=1000 Hilltop Circle, city=Baltimore,
postcode=21250, state=MD, country=USA

# A clear case for dust obscuration of the lunar retroreflectors

Sanchit Sabhlok<EMAIL_ADDRESS>James B. R. Battat  Nicholas R. Colmenares
Daniel P. Gonzales  Thomas W. Murphy  Jr  [  [  [  [  [

###### Abstract

The passive retroreflector arrays placed on the moon by Apollo 11, 14 and 15
astronauts continue to produce valuable Earth-Moon range measurements that
enable high-precision tests of gravitational physics, as well as studies of
geo- and selenophysics. The optical throughput of these retroreflectors has
declined since their deployment, with an additional signal loss at full moon
when the reflectors experience direct solar illumination. We show that the
loss in return rate can be attributed to the accumulation of a thin layer of
lunar dust on the surfaces of the corner cube retroreflectors. First, a
careful analysis of the optical link budget for the Apache Point Observatory
Lunar Laser-ranging Operation (APOLLO) experiment reveals that the lunar
return rate is 15–20 times smaller than predicted, a deficit that can be
explained by a reflector dust covering fraction of ${\sim}50$%. Second, range
measurements taken during a lunar eclipse indicate that the solar illumination
of the retroreflectors degrades their throughput by an additional factor of
${\sim}15$. Finally, a numerical simulation of heat transfer in dust-coated
reflectors is able to model the resulting thermal lensing effect, in which
thermal gradients in the retroreflectors degrade their far-field diffraction
pattern. A comparison of this simulation to eclipse measurements finds a dust
coverage fraction of ${\sim}50$%. Taken together, the link analysis, eclipse
observations and thermal modeling support the claim that the retroreflectors
are obscured by lunar dust, with both link budget and simulation independently
finding the dust fraction to be $\sim$50%.

###### keywords:

Eclipses Moon, surface Regoliths

##  1 Introduction

The corner cube reflectors (CCRs) placed on the moon by Apollo 11, 14 and 15
astronauts continue to produce scientific output more than 50 years after
their deployment. The reflectors were designed to sit passively in the
periodic temperature swings of the lunar environment in order to facilitate
Lunar Laser Ranging (LLR) measurements. It would be fair to say that these
reflectors have greatly outperformed expectations, continuing to operate for
more than 50 years after placement. Measurements of the Earth–Moon distance
provide precision tests of fundamental physics (e.g. general relativity,
Lorentz Invariance, time evolution of fundamental constants), as well as
geophysical information and constraints on the composition and dynamics of the
lunar interior (Murphy 2013).

The Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) began its
science campaign in 2006 (Murphy et al. 2008) with the goal of providing
millimeter-accuracy range data to improve constraints on gravitational
physics.

When APOLLO started ranging, it became clear that the measured return signal
was lower than expected by about a factor of 10 from careful link budget
calculations (Murphy et al. 2007). Surprisingly, the signal fell by an
additional order of magnitude when the lunar phase was within $\sim
20^{\circ}$ of full moon (Murphy et al. 2010). Murphy et al. (2010) discussed
various scenarios that could lead to the degradation of performance over four
decades in the lunar environment, the favored scenario being deposition of
dust on the retroreflector surfaces. In a subsequent work, Goodrow and Murphy
showed that thermal gradients of a few degrees in a CCR would lead to dramatic
suppression of the central intensity of the far-field diffraction pattern
(FFDP) Goodrow and Murphy (2012). While this does imply that the return signal
from the reflectors would be lowered as a result of a thermal gradient, the
paper did not perform any thermal modelling of the CCR under solar
illumination.

Lunar dust particles must rely on impacts and on electrostatic charges to
levitate and transport across the lunar surface due to a lack of atmosphere
(Colwell et al. 2007). Evidence for dust levitation was first seen in the
western horizon pictures taken by the Surveyor 5, 6 and 7 spacecraft (Criswell
1973, Rennilson and Criswell 1974) shortly after sunset. These images show a
distinct glow just above the horizon, indicating forward scattering of the
incident solar flux off a levitating dust cloud $<1$m from the ground. These
images compliment the reporting of streamers originating from the lunar
horizon as seen by the Apollo 17 astronauts from the spacecraft, indicating
the presence of levitating dust at scale heights of 5–20 km (McCoy and
Criswell 1974, Zook and McCoy 1991). Results from the Lunar Ejecta and
Meteorites Experiment (LEAM) placed on the moon by the Apollo 17 astronauts
indicated the presence of slow moving dust particles. These particles were not
detected uniformly over the course of a lunar day, but peaked strongly around
lunar sunrise, with detection rates increasing around 60 hours before sunrise
and persisting 30-60 hours after sunrise (Berg et al. 1976). The origin of
charge on lunar dust grains can be due to friction between meteorically-
agitated dust grains, photoelectric charging from incident solar radiation or
charging by the incident solar wind. The impact of solar radiation and wind on
the lunar surface creates a plasma sheath over the sunlit and shadowed regions
on the surface (Halekas et al. 2011). Experiments have verified that charged
dust grains can levitate in such a plasma sheath, allowing for their transport
across lunar surfaces (Arnas et al. 2001, Sickafoose et al. 2002).

In this work, we present a three-pronged approach to identify the cause of
reflector underperformance and lowered signal during the full moon. Firstly,
we perform a detailed link budget analysis in Section 2, supplemented by
observations of stars of known magnitudes and by beam scans across the lunar
reflectors to carefully analyze the profile of the transmitted laser beam. We
then compare these calculations to observed return rates from the moon, and
indeed find that the actual return rates are 15–20 times lower than expected.
Secondly, in Section 3, we describe results of APOLLO observations during
lunar eclipses—especially one in April 2014, under photometric
conditions—showing that the return rate improves by more than an order of
magnitude soon after a reflector enters the umbral shadow and cools. Section 4
presents results from our custom thermal simulation of a CCR in which we model
the expected lunar ranging return rate as a function of solar illumination
during a lunar eclipse. We discuss the implications of these results in
Section 5, and summarize our results in Section 6.

##  2 Link Budget

The laser used for APOLLO generates upwards of $10^{17}$ photons per pulse.
Under good conditions, only about one round-trip photon is detected per pulse.
A multitude of factors—dominated by divergence—contribute to such a high
attrition rate. This section evaluates the link budget, aiming to account for
all signal losses along the beam’s round-trip journey from the laser to the
detector by way of the moon. We will compare the results of the link equation,
Eq. 1, to observations on two high-performance epochs, adjusting some of the
parameters to suit changing conditions. The nominal link budget is:

$N_{\mathrm{detect}}=N_{\mathrm{launch}}\eta_{\mathrm{launch}}\eta_{\mathrm{c}}^{2}\eta_{\mathrm{r}}\eta_{\mathrm{NB}}Q\eta_{\mathrm{FOV}}\eta_{\mathrm{refl}}N_{\mathrm{refl}}p_{\mathrm{gauss}}\bigg{(}\frac{d}{r\phi}\bigg{)}^{2}p_{\mathrm{refl}}\bigg{(}\frac{D_{\mathrm{eff}}}{r\Phi}\bigg{)}^{2}\prod_{i}D_{i}.$
(1)

Here, the number of photons we expect to detect is denoted as
$N_{\mathrm{detect}}$. Table 1 elucidates the meaning of the terms on the
right, elaborated in the text.

Table 1: Parameters used in link equation (Eqn. 1). Values for $\eta_{\mathrm{r}}$ and $Q$ are not determined separately, using the measured amalgam $\eta_{\mathrm{c}}\eta_{\mathrm{r}}\eta_{\mathrm{NB}}Q$ instead. Blank uncertainties are either meaningless or too small to matter. Some parameters are adjusted for specific observations. Parameter | Description | Value | Uncertainty
---|---|---|---
$N_{\mathrm{launch}}$ | Laser photons per pulse | $2.7\times 10^{17}$ | $0.14\times 10^{17}$
$\eta_{\mathrm{launch}}$ | Central obscuration in uplink | 0.60 | 0.03
$\eta_{\mathrm{c}}$ | Common-path optical throughput | 0.53 | 0.09
$\eta_{\mathrm{r}}$ | Receiver efficiency | — | —
$\eta_{\mathrm{NB}}$ | Narrow-band filter transmission | 0.99 | 0.003
$Q$ | Photon detection efficiency | — | —
$\eta_{\mathrm{c}}\eta_{\mathrm{r}}\eta_{\mathrm{NB}}Q$ | One-way throughput amalgam | 0.053 | 0.007
$\eta_{\mathrm{FOV}}$ | Limited detector field of view | 0.787 | adjusted
$\eta_{\mathrm{refl}}$ | Corner cube optical transmission | 0.93 | 0.005
$N_{\mathrm{refl}}$ | Number of reflectors in array | 100 or 300 | —
$p_{\mathrm{gauss}}$ | Gaussian profile vs. top-hat | 0.693 | —
$d$ | Diameter of single corner cube | 0.0381 m | —
$r$ | Earth-Moon distance |  $3.85\times 10^{8}$ m | adjusted
$\phi$ | Uplink beam divergence | 1.0 ′′ | adjusted
$p_{\mathrm{refl}}$ | TIR CCR diffraction vs. top-hat | 0.182 | —
$D_{\mathrm{eff}}$ | Effective telescope aperture | 3.26 m | —
$\Phi$ | Downlink beam divergence | 2.89 ′′ | —
$D_{i}$ | Adjustments for epoch particulars | Table 2 | —

APOLLO’s laser operates at a wavelength $\lambda=532$ nm, and the energy per
pulse of the laser is $\sim 0.1\mathrm{\,J}$, resulting in
$N_{\mathrm{launch}}\approx 2.7\times 10^{17}$ generated photons per pulse.
Transmission losses are accounted later, but we separately address the
geometrical loss of the outgoing Gaussian beam due to the secondary mirror
obstruction via $\eta_{\mathrm{launch}}=0.60\pm 0.03$.

Various transmission and reflective efficiencies associated with specific
segments of the photons’ overall path are captured by $\eta$ factors in Eq. 1.
The first efficiency, $\eta_{\mathrm{c}}$, is the one-way optical efficiency
that is common to both the transmit and receive paths of the APOLLO
system—including atmospheric losses. For Apache Point, $\eta_{c}$ evaluates to
$0.53\pm 0.09$ at 532 nm under clear skies and pointing at zenith (three
reflections at $0.85\pm 0.05$ and 0.87 atmospheric transmission). Because this
path is traversed twice—once for the transmitted beam and once for the
received signal—$\eta_{\mathrm{c}}$ appears squared in Eq. 1. The combined
term $\eta_{\mathrm{c}}\eta_{\mathrm{r}}\eta_{\mathrm{NB}}Q\approx 0.053$
relates to the total one-way throughput of the receiver/detector, which is
experimentally determined as a group in Section 2.2. The $\eta_{\mathrm{FOV}}$
term, nominally computed for a 1.0 arcsec seeing disk, represents the fraction
of the point spread function (PSF) captured by the 1.4 arcsec square field of
view of the APOLLO receiver, with one corner pixel in the $4\times 4$ array
detector unused.

The corner cube reflectors (CCRs) number $N_{\mathrm{refl}}$ in the reflector
assembly (100 for Apollo 11 and 14; 300 for Apollo 15), have a front-surface
reflection loss represented by $\eta_{\mathrm{refl}}$, and diameter, $d$.

The first squared term in parentheses represents the fraction of the uplink
beam reaching a single CCR, according to a top-hat profile having a nominal
diameter of $\phi=1.0$ arcesc (see Section 2.1). For the divergence geometry,
the one-way telescope-reflector distance is set to the average Earth–Moon
distance of $r=3.85\times 10^{8}$ m. We later correct for the geometry at the
epoch of observation via derating terms, $D_{i}$. The erroneous top-hat model
is then corrected to describe a Gaussian profile more characteristic of
atmospheric seeing, whose full-width at half-maximum (FWHM) matches the
diameter of the top-hat profile. The correction factor,
$p_{\mathrm{gauss}}=\ln 2$, accounts for the reduced central intensity of a
Gaussian profile having the same total flux and FWHM as the top-hat (FWHM is
$\sqrt{\ln 256}$ times the Gaussian sigma).

The second squared term in parentheses captures geometric loss in the
returning beam, whose divergence is also initially treated as a top-hat
illumination profile of angular diameter $\Phi=\lambda/d$. This is intercepted
by a telescope whose effective diameter is such that the collecting area is
$\pi D^{2}_{\mathrm{eff}}/4$. The top-hat model is then corrected to represent
the central intensity of a far-field diffraction pattern arising from total
internal reflection (TIR) by the parameter $p_{\mathrm{refl}}$ Murphy and
Goodrow (2013).

Note that while not appearing explicitly in Table 1, a characteristic
uncertainty of 0.05 arcsec is applied for $\phi$, which propagates into an
uncertainty value for $\eta_{\mathrm{FOV}}$ of 0.03. These uncertainties are
applied with the rest in Section 2.4.

###  2.1 Beam Scan Test: Lunar Beam Profile

In order to assess the uplink divergence, $\phi$, we contrive to scan the
telescope-fixed laser across the reflector while keeping the detector centered
on the reflector. This requires a coordinated move of the receiver’s
adjustable offset pointing angle to counter the motion of the telescope (and
thus outgoing beam). We perform a scan in two dimensions, effectively in a
plus-shaped pattern centered on the nominal pointing. As each pixel in the
$4\times 4$ detector only spans 0.35 arcsec, very small motions are
noticeable.

As such, a scan of this sort is easily thwarted by pointing drift. Moreover,
even when maintaining the pointing to the best of our ability, factor-of-two
variations in signal on the timescale of tens of seconds are routine, largely
due to seeing variations (APOLLO is highly sensitive to seeing, both due to
target illumination and overfilling of the small detector). This makes
interpreting a beam scan tricky: how much of the variation is due to the
pointing offset, how much is variable seeing, and how much is due to variation
in other terms in the link budget?

To mitigate this, we execute a pattern of moves that returns to the central
position after every offset, thus interleaving central positions between each
excursion. During the times pointing at the nominal (central) position, we
sometimes adjust the pointing to re-center and maximize signal. Furthermore,
we only consider the signal change in the several seconds surrounding the
transition from one position to the next, rather than averaging the whole
offset result. Each pointing therefore gives two signal ratios: at the
beginning of the move, and again at the end.

Fig. 1 shows an example from one epoch, in orthogonal dimensions. Each data
point averages the beginning and end ratios for the excursion. Since all
measurements are in relation to the return rate at the central position, the
result is normalized to the (changing) central value for each paired
measurement.

Figure 1: Beam scan results on 2018 February 25 for $x$ (blue circles) and $y$
(red triangles) directions, normalized to the peak. The nominal center
positions are denoted by stars. In this case, the FWHM of the Gaussian fits
are $1.001\pm 0.06$ and $1.004\pm 0.04$ arcseconds.

We report here results from beam scans on two epochs. For 2018 February 25, we
measure an uplink beam FWHM of $1.00\pm 0.05$ arcsec, while on 2018 September
28 we get $1.07\pm 0.19$ arcsec. Because the $\phi$ parameter in Eq. 1 is set
to 1.0 arcsec, and squared in the link equation, we apply a derating factor of
0.88 to the latter case (the former needing no correction), as will be
summarized in Section 2.3. The distribution of lunar return photons on the
receiver array further corroborates—and closely matches—the beam scan results
by providing an in-situ characterization of the seeing at the time of
observation.

###  2.2 One-Way Throughput

A number of the terms in Eq. 1 can be simultaneously determined by pointing to
a non-variable bright star of known magnitude and spectral type, measuring the
photon counts, then comparing to theoretical expectations. We first assess the
top-of-atmosphere flux density at 532 nm (in W m-2 nm-1, e.g.) for the
spectral type and magnitude, correct for the zenith angle, sum over the
effective telescope aperture, multiply by the 2.1 nm bandwidth of the narrow-
band filter, and by the effective integration time (typically 500 exposures at
100 ns each), and divide by photon energy to arrive at the number of photons
we would detect in the case of no losses. By comparing actual counts, we
effectively measure the product
$\eta_{\mathrm{c}}\eta_{\mathrm{r}}\eta_{\mathrm{NB}}Q\eta_{\mathrm{FOV}}$.

In practice, we fit a two-dimensional symmetric Gaussian profile to each 50
$\mu$s exposure (typically delivering about 1,000 counts across 15 active
elements) to determine the theoretical total flux: $2\pi A\sigma^{2}$, where
$A$ is Gaussian amplitude and $\sigma$ is the standard deviation width
parameter—if unconstrained by $\eta_{\mathrm{FOV}}$. This produces a more
stable flux estimate than the variable, jumpy seeing produces in raw
terms—since the fraction of light captured in the detector is sensitive to
offset and width of the PSF in a way that the total flux is not.

Several such tests on 2019 July 20 under the same detector configuration as
for the beam scan epochs resulted in a determination for
$\eta_{\mathrm{c}}\eta_{\mathrm{r}}\eta_{\mathrm{NB}}Q$ of $0.053\pm 0.007$.
We need only multiply by another factor of $\eta_{\mathrm{c}}$ to represent a
cluster of terms in Eq. 1. The term $\eta_{\mathrm{FOV}}$ can be computed
exactly for a centered Gaussian in a $4\times 4$ grid whose pixels are 0.35
arcsec and missing one corner. For example, a Gaussian whose FWHM is 1.0
arcsec delivers 78.7% of its flux to such an array, while 1.5 arcsec seeing
would result in 50.6%. We use the 1.0 arcsec value in Eq. 1, then correct for
conditions in Section 2.3.

###  2.3 Derating Factors

Some of the parameters in Eq. 1 are not constant, and require adjustment for
the time of observation. Moreover, a few additional factors impact the link
and need to be considered. These terms appear at the end of Eq. 1 as a product
of derating factors. The various derating factors applied in this analysis
appear in Table 2.

Table 2: Link budget derating factors for the two beam profile experiments.  Cause | Epoch A Value | Derating | Epoch B Value | Derating
---|---|---|---|---
| 2018 Feb 25 |  | 2018 Sep 28 |
Range ($r$) |  $3.6008\times 10^{8}$ m | 1.31 |  $3.7724\times 10^{8}$ m | 1.08
Zenith Angle | $19.9^{\circ}$ | 0.98 | $47.8^{\circ}$ | 0.87
Angular Offset (libration) | $5.7^{\circ}$ | 0.75 | $8.13^{\circ}$ | 0.66
Sun Angle ($\theta$) | $-67.3^{\circ}$ | 0.84 | $40.8^{\circ}$ | 0.74
Velocity Aberration | 0.946 ′′ | 0.762 | 1.003 ′′ | 0.736
Uplink Beam ($\phi$) | 1.00 ′′ | 1.0 | 1.066 ′′ | 0.88
Detector Capture ($\eta_{\mathrm{FOV}}$) | 0.90 ′′ | 1.05 | 0.91 ′′ | 0.950
Total Derating |  | 0.648 |  | 0.282

First, the Earth–Moon distance varies by as much as 13% (full range), and
appears to the fourth power in Eq. 1. Based on the distance,
$r_{\mathrm{obs}}$, at the time of observation, we apply a “derating” factor
(that indeed enhances the link result as often as not) of
$(r/r_{\mathrm{obs}})^{4}$.

The second factor accounts for the fact that we do not operate at zenith
pointing, and is provided by the expression $0.87^{2(\sec z-1)}$. Here, $z$ is
the zenith angle of the Moon. The base of 0.87 comes from the atmospheric loss
by passing through one atmosphere at 532 nm. The factor of two in the exponent
is to account for the beam passing through the atmosphere twice. Finally, we
subtract one in the exponent because we already account for one atmosphere of
thickness in $\eta_{\mathrm{c}}$.

Three other derating factors come from the geometry of the Earth–Moon and
Moon–Sun relationships. First, libration of the moon results in an angular
offset of the CCR array relative to the Earth–Moon line, producing a reduction
of the central irradiance of the far-field diffraction pattern via reduced
area and increased diffractive spread. Next, thermal computations by the
reflector manufacturer Faller et al. (1972) anticipated modest dependence of
reflector behavior on sun angle $(\theta)$. Finally, velocity aberration
shifts the return far-field diffraction pattern on the surface of the Earth by
approximately one arcsecond, so that the telescope sits not at the peak of the
pattern, but on the dimmer shoulder. The central lobe of the complex TIR
diffraction pattern closely follows that of an Airy pattern from a circular
aperture of the CCR’s diameter Murphy and Goodrow (2013). The derating amount
is thereby computed using this profile based on the relative tangential
velocity at the time of observation—dominated by Earth rotation.

Lastly, atmospheric seeing impacts both the outgoing beam divergence ($\phi$),
and the fraction of light captured by the detector, $\eta_{\mathrm{FOV}}$. The
latter is calculated according to a Gaussian fit to the lunar return
distribution on the $4\times 4$ detector, whose FWHM is given in Table 2,
along with the corresponding adjustment relative to the nominal case presented
in Table 1. It is reassuring that the FWHM values for the beam scans are not
much larger than the FWHM values of the lunar signal on the
detector—indicating good beam collimation so that the outgoing beam is
essentially seeing-limited. Note that on 2018 September 28, an additional
detector element near the center was inoperative, which is why the derating
factors are different, despite similar FWHM measures for the two cases.

###  2.4 Link Assessment and Implied Dust Fraction

The previous sections laid the groundwork for comparing observed lunar return
rates to theoretical expectations. Application of Eq. 1 using values in Table
1 and Section 2.2 produces for Apollo 15 a nominal expectation of $19.1\pm
4.8$ photons per shot, before derating factors are applied. For the two epochs
considered, the derating factors yield approximately 12.4 and 5.4 photons per
shot.

On these two nights—the ones for which beam scans were performed, thus
obtaining confident optimization of the signal—we assess the photon return
rate at the many visits to the central beam position. Because the LLR signal
rate is extremely sensitive to telescope pointing and atmospheric seeing (via
$\phi$ and $\eta_{\mathrm{FOV}}$ parameters), we see much variation in
central-position rates throughout the roughly ten-minute scan procedure. We
therefore take special care in assessing a “reasonable” return rate, looking
to identify a consistent “best” central rate that recurs multiple times
throughout the scan and that is therefore compatible with the beam divergence
assessed over the entire scan period. We believe these “best” rates are more
representative of the theoretical link performance of APOLLO, for more
meaningful comparison to the link calculation.

The estimated return rates for the two epochs are assessed to be 0.66 and 0.35
photons per shot in February and September of 2018, respectively. The observed
rates are therefore well below the theoretical expectations of roughly 12 and
5 photons per shot. The ratios are similar, at 5.2% and 6.7% for the
respective epochs. It should be noted that the derating factors produce
expectations roughly a factor-of-two different for the two epochs, and that
the observed rates are likewise different by approximately the same
factor—indicating consistent results.

We therefore conclude that observed return rates are at least an order-of-
magnitude lower than expectations, and are unable to account for the
difference by careful analysis and experimental characterization. A proposed
explanation that runs through this paper, and others before it Murphy et al.
(2010, 2014), is that dust accumulation on the front surfaces of the CCRs
simultaneously accounts for the link deficit, the additional full-moon
deficit, and eclipse response.

Treating the dust as a set of geometrical obstructions, we recognize that the
double-pass nature of the CCR makes each dust grain count twice, blocking both
entering and exiting light paths. If the geometrical covering fraction is $f$,
the probability of passage through the surface is $1-f$. Independent
probabilities for blockage in the two directions means that the probability of
round-trip passage is $(1-f)^{2}$. This amount of blockage will impact the
central intensity of the far-field diffraction pattern (FFDP). Normally, the
unobstructed CCR, with front surface area, $A$, will have a FFDP central
intensity proportional to $A^{2}$ (effectively, a coherent sum of the electric
field over the open aperture, squared for intensity). Since the obstructed CCR
has an effective area of $(1-f)^{2}A$, the FFDP central intensity will be
reduced by $(1-f)^{4}$. This strong dependence on $f$ means that, for
instance, a 50% covering fraction results in a sixteen-fold reduction in the
FFDP central intensity—which is indeed approximately the factor of signal
deficit we deduce. Applied to the signal deficit ratios from the two epochs
above, we compute dust fractions of 0.52 and 0.49 for February and September
2018, respectively. The two epochs are therefore telling a very similar story
in the context of dust coverage.

We note that infrared observations may be less impacted by dust, as the
particles may be less “geometrical” at longer wavelengths. This may be related
to the success of recent infrared LLR operations, where reflector performance
appears to be better (Courde et al. 2017). It is also the case the thermal
lensing effect explored more below is less impactful at longer wavelengths.

##  3 Eclipse Observations

Figure 2: (Top) Histogram of the lunar return rate (photons per shot) during
four eclipses (black line), as well as for the photometric eclipse on April
15, 2014 alone (red). Also shown (blue fill) is data from the photometric
eclipse when the target reflectors are in full shadow. Note the change of
scale on the ordinate indicated by the horizontal dotted line. (Bottom) Lunar
return rate histogram when the moon is not in eclipse, but the lunar phase
angle is close to full (within 10 degrees). All non-eclipse lunar return rates
are less than 0.2 photons per shot. Both histograms aggregate data from all
three Apollo reflector arrays (Apollo 11, 14 and 15). The Apollo 11 and 14
rates have been scaled up by a factor of three because those arrays contain
three times fewer corner cube reflectors than the Apollo 15 array.

To study the effect of solar illumination on ranging throughput, we took
advantage of the naturally occurring variation in the solar illumination
incident on CCRs during lunar eclipses. During a lunar eclipse, the solar
illumination changes from 100% to 0% (shadow) and back to 100% over the course
of about five hours as the CCRs enter and then exit the Earth’s umbral shadow.
A measurement of the evolution of the lunar return rate during the eclipse
provides an independent constraint on the CCR dust coverage fraction, which
can be compared to the fractions derived using link budget calculations
(Section 2.4) and thermal eclipse simulations (later in Section 4).

We undertook four lunar eclipse ranging campaigns (December 21, 2010; April
15, 2014; September 28, 2015; January 21, 2019). Here, we focus on the 2014
eclipse (MJD 56762 from 5:52 to 10:52 UTC) because photometric sky conditions
that night enabled the most reliable study of the throughput of the APOLLO
system and lunar retroreflectors. Range measurements were divided into short
runs of $\sim$150 second duration (3000 laser shots) in order to monitor the
evolving lunar return rate as a function of solar illumination of the lunar
retroreflectors.

We compare eclipse-night observations to lunar return rates measured during
non-eclipse nights, but with the lunar phase within 10 degrees of full moon.
In those cases, the median (mean) return rate is 0.02 (0.03) photons per shot,
and the maximum is 0.2 photons per shot (see the bottom plot of Fig. 2).
During the 2014 eclipse, before the Apollo 11 and 14 reflectors entered the
earth’s umbral shadow the median (mean) return rate was also low, at 0.06
(0.08) photons per shot (red histogram in the upper plot of Fig. 2). After the
reflectors entered the umbral shadow and began to cool (blue shaded histogram
in the upper plot of Fig. 2), the return rate rose as high as 0.77 photons per
shot, with a median (mean) value of 0.35 (0.41) photons per shot, which is 17
times higher than the median rate during non-eclipse nights near full moon.
The lunar return rate histogram for data from all four eclipses is shown in
black in the top plot of Fig. 2, and it tells a consistent story: under
overhead solar illumination, the optical throughput of the lunar reflectors
degrades by at least an order of magnitude, but when the illumination is
suppressed (via a lunar eclipse) and the reflectors cool, the rate increases.
As described more fully in Section 4.2, we attribute the rate increase to the
cooling of the reflectors when they are in shadow, which in turn reduces the
thermal gradients in the CCR and improves the FFDP (and therefore the optical
throughput) of the corner cubes.

##  4 Thermal Modelling of the Corner Cube Retroreflectors

Figure 3: (Left) Schematic diagram defining the geometry of the CCR mounted in
the aluminum casing (”shell”) via Teflon mounting rings and the retainer ring.
(Right) Elements and conductive paths used in our thermal simulation, with
specified thermal conductances. The CCR, retainer ring, and the shell radiate
heat to space. Solar energy is incident on the top surfaces of the CCR,
retainer ring, and shell. The corner cube looks asymmetric from this angle
because the tabs are placed at 120 degree intervals around its perimeter, so
diametrically opposite sides do not both have tabs.

We model the optical response of a CCR under thermal load to estimate the dust
coverage fraction for comparison with the degradation estimate from our link
budget calculation in Section 2, the eclipse observations in Section 3, and
the known degradation of performance during full moon (Murphy et al. 2010).
Our simulations were done using a custom code in C developed by our group for
this purpose. The simulation has three stages. First, we set up the geometry
of the problem, defining the different surfaces, their thermal properties and
initial conditions. In the second stage, the code calculates the analytic view
factors from all boundary point surface elements to all others and stores them
into a view factor matrix to be referenced later for radiative heat transfer.
Lastly, the code performs conductive and radiative heat transfer, based on the
initial conditions for a specified amount of time. The details of the code
implementation, including the geometry setup, view factor matrix construction
and heat transfer, are described in Appendix A.

A schematic diagram and corresponding thermal model of the mounted CCR is
shown in Fig. 3. The fused silica CCR is mounted to an aluminum shell
(aluminum 6061-T6), held in place by two teflon rings and an aluminum retainer
ring (anodised aluminum 1100). For the CCR, retainer ring and the shell, we
have to specify the net absorptivity and emissivity as a result of an impeding
dust fraction sitting on the surfaces. To define the net emissivity and
absorptivity, we consider a fraction $f$ of the surface to be obscured by
lunar dust, with an inherent emissivity $\epsilon_{dust}$ and absorptivity
$\alpha_{dust}$. For the shell and the retainer ring, the net emissivity is
simply an area-weighted mean of the two emissivities. Effective emissivity and
absorptivity for the top of the shell and retainer ring is given by ϵ_eff =
(1-f) ϵ\+ f ϵ_dust
α_eff = (1-f) α\+ f α_dust. The case for the top of the CCR is slightly more
complicated by the fact that light can be absorbed by the dust on the way back
out of the CCR as well. So the effective absorptivity comes out to be α_eff,
CC = [1 - (1-f)^2] α_dust.

Our simulation investigates both equilibrium and dynamic modes. First, a
constant heat input, modelling the solar flux incident on the moon (1360 W
m-2), is used to simulate an equilibrated CCR temperature profile under full-
moon illumination conditions. As our interest in this work primarily concerns
observations near full moon, we do not consider the possibility of sunlight
illuminating the interior of the shell behind the CCR because all incident
sunlight is rejected by total internal reflection when the solar illumination
is within 17 degrees of normal incidence. The equilibrated temperature profile
is then used as a starting point for an eclipse simulation, where the solar
flux is varied according to an eclipse illumination profile. The simulation
computes the temperature of all elements at each time step, from which the
temperature profile of the CCR is extracted.

Given the CCR temperature profile, the path length variations imposed by
thermal modification of refractive index within the CCR are calculated,
leading to a warped wavefront emerging from the CCR to produce a FFDP—the so-
called thermal lensing effect Goodrow and Murphy (2012). The effect is largely
driven by the temperature gradient from the surface to the corner of the CCR.
The resultant FFDP can then be compared to the case of an isothermal CCR, to
quantify the loss in return rate as a percentage. We calculate this loss for
the full moon illumination, which can be compared to our measured return rate
during full moon observations (Murphy et al. 2010). We also compare the
modeled return rate improvement during the eclipse to that observed during the
2014 eclipse. The thermal properties for the different materials used in the
simulation are outlined in Table 3.

Table 3: Properties of materials used in the thermal simulation.  Element | Material | Thermal Conductivity | Density | Heat Capacity | Emissivity | Absorptivity
---|---|---|---|---|---|---
|  | $\rm{W}\,\rm{m}^{-1}\,\rm{K}^{-1}$ | $\rm{kg}\,\rm{m}^{-3}$ | $\rm{J}\,\rm{kg}^{-1}\,\rm{K}^{-1}$ |  |
CCR | Fused Silica | 1.38 | 2200 | 740 | 0.87 | 0.0
Rings 1 & 2 | Teflon | 0.25 | 2200 | 1500 | 0.8 | 0.8
Shell | Aluminum 6061-T6 | 167.0 | 2700 | 896 | 0.025 | 0.06
Retainer Ring | Anodized Aluminum 1100 | 222.0 | 2710 | 904 | 0.78 | 0.32
| Lunar Dust | — | — | — | 0.9 | 0.9

###  4.1 Steady-state Simulation Results

To compute the steady-state thermal profile of the CCR in the presence of
direct overhead solar illumination (as would be the case during a full moon),
we begin the simulation with all elements set to a temperature of 300 K, and
assume a solar constant value of 1360 W m-2. We then run the simulation until
the average temperature of all four simulated elements (CCR, Teflon rings,
retainer rings and shell) stabilize.

The spatial temperature profile of the CCR is extracted and used to generate a
FFDP for the CCR. Different values of $f$ result in differing central maximum
intensities compared to the isothermal CCR. We find that varying $f$ from 0.5
to 0.65 changes the central maximum intensities of the CCR from ${\sim}6$% to
about ${\sim}0.1$% of the isothermal case. The peak-to-peak phase difference
in the wavefront at a dust fraction of 0.5 is $\sim$ 4.9 radians due to
thermal effects. This phase difference across the wavefront results in a
diminished central lobe of the FFDP. The former result, at $f=0.5$, matches
the full-moon deficit factor of 10–15 reported in Murphy et al. (2010), and
the results from Section 3 and Fig. 7 of this paper.

It is worth briefly discussing the qualitative thermal behavior of the CCR
when we simulate equilibrating the geometry for a constant solar flux. As the
run is started, a thermal gradient is set up quite rapidly within the CCR,
within the first 100 seconds of temporal evolution. The surface of the CCR
absorbs heat from incident solar radiation. This creates the thermal gradient
in the CCR, with the front face being hotter than the vertex. This thermal
gradient is responsible for the reduced intensity in the central maximum of
the FFDP. It takes far longer, a few hours, for the system to reach a steady-
state temperature. The main cooling channel for the system at this point is
the radiative cooling from the CCR front surface, radiating heat to space. We
confirmed that the final steady-state temperatures are independent of initial
conditions. Since the thermal gradients within the CCR rapidly adjust to the
incident solar flux, we expect that during the eclipse simulation, the return
rate should quickly improve once the reflector enters the shadow, since the
gradients, rather than the absolute temperature, affect the FFDP and therefore
the return rate. This is indeed the case as shown in the next subsection.

###  4.2 Simulation Result Showing Eclipse Profile

We apply our simulation to the case of a lunar eclipse in which the solar
illumination incident on the CCRs is modulated by the Earth’s shadow. The
relative angular sizes of the Earth and Sun as seen from the moon are used to
simulate an eclipse profile for different impact parameters for the Earth’s
center passing in front of the Sun. This is compared to the illumination
profile seen at the three different Apollo reflectors Chapront-Touzé and
Chapront (1991). The computed illumination profile does not include limb
darkening. The impact parameter that best describes the profile for all three
reflectors is used as the incident solar flux profile, as shown in Fig. 4.
While all reflectors remain under the shadow for different amounts of time,
the total time difference between the shortest and longest eclipse times for
different reflectors is approximately 6–7 minutes. When comparing to APOLLO
observations, this is shorter than the circuit among reflectors, so the impact
of different reflectors being under the shadow for different amount of times
is minimized. The resultant temperatures for the CCR, teflon rings, the
retainer ring and the shell are shown in Fig. 5. Since the FFDP evolution is
largely driven by the corner cube gradients, we present the radial and
vertical gradients within the CCR in Fig. 6. For the radial gradient, due to
the trifold symmetry of the corner cube, we consider a radial gradient between
the center and a point on the circumference which is a midpoint of one of the
“tabs.” We also consider a point on the circumference equidistant from the two
nearest tabs for a radial gradient.

The qualitative behavior of the system during the eclipse can be described as
follows. Starting from an equilibrated geometry, as the incident solar flux
gradually decreases, the amount of heat absorbed by the CCR decreases as well.
According to Figure 6, very little radial gradient is initially present, and
thus little conduction through the tabs (the chell and CCR being at a similar
temperature). Consequently, the CCR loses thermal energy primarily through
radiation from the front surface. Before long, the CCR becomes cooler than the
shell, so heat conducts into the CCR through the mounting tabs, setting up
radial gradients while maintaining a weak vertical gradient as shown in
Figures 5 and 6. When the solar flux dwells at zero, the CCR continues to
cool, but the gradient is relatively stable, thus resulting in the plateau
region in the expected return rate shown in Figure 7. When the reflectors come
out of the shadow, solar flux increases so that the CCR starts to absorb more
heat and the strong vertical gradient develops again, depressing the expected
return rate. Throughout this process, the CCR front surface remains hotter
than the vertex, i.e. the gradient maintains the same orientation.

The time dependence of the ratio between the intensity of the central maximum
in the FFDP of the CCR during an eclipse with respect to an isothermal corner
cube is plotted in Fig. 7, along with the measured return rates from
individual Apollo reflectors during the 2014 eclipse. The simulated return
rate increases rapidly after the CCR enters the Earth’s shadow, and is
correlated with a rapid reduction in the temperature gradient from the back to
the front of the CCR as it cools as shown in Fig. 6. As the CCR stays in the
shadow, it retains the low vertical gradient compared to steady state
illumination during full moon. When the reflectors come out of the shadow, the
gradient increases to pre-eclipse levels rapidly, destroying the central
maximum in the FFDP resulting in low return rates from the CCR. The simulated
ratio between the maximum intensity in the FFDP during the eclipse to the
start of eclipse is ${\sim}7$, as the intensity goes from ${\sim}$6% to
${\sim}42$%. This is comparable to the observed ratio of median return rate
(0.35) during an eclipse with respect to the original rate (0.06), i.e. a
factor of 6, as shown in Fig. 7 and reported in Section 4.2. Return rate
observations in Fig. 7 show a low return rate prior to eclipse, followed by a
rapid rise in the return rate (a factor of ${\sim}7$ in 10000 seconds) after
the CCRs enter the umbral shadow.

Our simulation includes only one CCR, and therefore will not capture all
elements present in the actual reflector array on the lunar surface. However,
our objective is to show that (1) dust reduces the intensity of the central
maximum in the FFDP of the CCR and (2) as the incident solar flux on the CCR
decreases during the lunar eclipse, the temperature gradient between front
face and the vertex rapidly decreases, improving the return rates.
Discrepancies between our simulation and the eclipse-night observations are
discussed in Section 5.

Figure 4: Percentage of maximum solar illumination as a function of time (in
seconds) for the three Apollo reflectors Apollo 11 (red), 14 (blue) and 15
(green) for the eclipse on April 15th, 2014, shifted to have the same starting
time. The blue dashed curve shows the eclipse profile corresponding to an
impact parameter of 1.75 solar angular radii as seen from the moon, which we
used in our thermal eclipse simulation since it best approximates the behavior
of all three reflectors. Note that these models do not include limb darkening.
Figure 5: Average temperatures for the CCR, Teflon rings, retainer ring and
shell over the course of the simulated eclipse. Dark shading indicates
$1\sigma$ deviations from the mean, whereas the lighter shading indicates the
min-max. The deviations for the shell and the retainer ring are too small to
be seen here due to their high thermal conductivity. The range of computed
temperatures for the Teflon rings during the simulated eclipse is larger than
for the other three elements. This is due to the fact that the temperatures
extracted from the simulation average the two Teflon rings that are above and
below the CCR. The rings also have slightly different conduction pathways to
carry heat from the CCR to the shell or the retainer rings. Some of these
pathways have an additional temperature dependence due to the discontinuity in
thermal conductance at the Teflon-Retainer Ring and Teflon-shell interface.
All of these effects combine to result in a larger range of temperatures for
the Teflon rings. Figure 6: Temperature differences between three pairs of
locations on the corner cube as a function of time during an eclipse.
Corresponding solar illumination profile is provided in Fig. 4. As the CCR
enters the umbral shadow it cools and the vertical temperature gradient drops
from $\sim$5 K to $\sim$2 K, which reduces the thermal lensing and improves
the optical throughput. The inset shows the front face of the CCR illustrating
locations used for the temperature difference calculations, the center (A),
tab midpoint (B) and a point between two tabs along the circumference (C). The
dashed lines connecting the points correspond to the gradients shown on the
plot. The vertical gradient is calculated by subtracting the temperature of
the vertex (D, not shown in inset) from the center of the top surface (A - D).
The radial gradient for the top surface is calculated for two points along the
top surface circumference, the midpoint of the CCR tabs (A - B), and the
midpoint between two tabs (A - C).  Figure 7: Fraction of light in the central
diffraction maximum as compared to an isothermal CCR as a function of time
during an eclipse. The measured return rates from the Apollo reflectors are
also plotted (right-hand ordinate). Time is relative to when each reflector
first enters the Earth shadow, which aligns all reflector time-series with the
start of the eclipse simulation. Simulated CCR FFDPs are shown at five
representative times during the eclipse. Inset at top left shows the FFDP for
an isothermal CCR. Figure 8: Simulated return rate fractions for the same dust
fractions (f$=$0.5), but with the original and rescaled (x10) mount
conductances (MC). The increased mount conductance almost completely depresses
the intensity of the central maximum in the FFDP throughout the eclipse, and
is therefore disfavored as inconsistent with observations.

##  5 Discussion

While the qualitative results from our eclipse simulation are in agreement
with the measured return rate during the photometric eclipse night, there are
a few details worth expanding on.

###  5.1 Simulating a Single CCR

We consider the effect of using a single CCR versus an entire array set in a
pallet. We model a single dust-obscured CCR to simulate its FFDP, and compare
the intensity in the central diffraction maximum to that of an isothermal CCR
as an estimate of the expected relative lunar return rate. The thermal and
mechanical properties implemented in this simulation have been either taken
directly or derived from the description in the original Arthur D. Little
Report on the CCR design and performance Arthur D. Little, Inc. (1969).
However, our single cylindrical shell and CCR is not likely to track the
thermal mass and environmental couplings of the real array, which could result
in incorrect temporal behavior. We did not model the changing radiative
temperature of the lunar surface, the geometry of coupling, or the blankets in
which the arrays are wrapped. We have attempted to approximate some of these
factors, but surely these are not perfect.

###  5.2 The Effect of Mount Conductances

Our goal was to model the thermal properties of the CCR using the
specifications indicated in the ADL design report Arthur D. Little, Inc.
(1969). A complication, however, was that the report doesn’t describe the
individual contact conductances but rather the total conductances through
various thermal pathways. The original report uses the term ”Mount
Conductance” to describe the total heat conductance from outside the shell
into the surface and vertex of the corner cube. To convert mount conductance
given in the ADL report into contact resistances used in the simulation, we
use the fact that the conductance values in ADL report are based on tests
using steady state heat flux from the corner cube, through the teflon rings,
through the retainer ring into the aluminum array which we have called the
”shell” in our simulation. In the simulation, the conductance values for
various interfaces between different materials can be tuned, however, the
caveat is that the teflon-retainer ring interface is a special case. Since the
retainer ring was tightened and then de-rotated by 36∘ at the time of assembly
to prevent any thermal contact, it is likely to approximate a no-contact
conductance. If this conductance value has changed since launch, it could only
be due to physical contact between the two surfaces, which is likely to change
conductance at this interface significantly. Thus, while our simulation can
change conductance values smoothly, for the real retroreflectors, such a
change will be discrete and significant. The computed conductance values for
this interface were checked by calculating two extreme cases. First, we
calculate the conductance for the case where there was perfect contact between
teflon and the retainer ring and then we calculate the case where there is no
contact between the retainer ring and the teflon ring and any thermal contact
is purely radiative. Our computed conductance value lies between these two
extremes, consistent with the value derived from steady state heat flux
assumption based on the ADL report. We calculate that the conductance for
perfect contact is approximately a factor of 10 larger than what we calculate
for the APOLLO CCRs. Thus, we are satisfied that the conductance values used
between the various interfaces are reasonable.

However, to study the possibility that the conductance between the teflon-
retainer ring surface has increased drastically, we consider the special case
of an eclipse where the mount conductance has increased by a factor of 10 as
shown in Fig. 8. The mount conductance values only impact our simulation if
they have increased since fabrication, since a decrease would result in
smaller gradients in the CCR, in agreement with our simulation and the
original predictions and measurements in the ADL report. In order to run a
full eclipse simulation, the modified geometry with increased mount
conductances is equilibrated. The initial temperature of the shell in this
case was found to be higher than the eclipse run presented in Fig. 5, by
$\sim$20 K. The expected intensity inside the central diffraction maximum is
decreased approximately by a factor of 40 during the eclipse compared to the
best match simulation with dust fraction of 0.5. Thus, adjusting the mount
conductances completely destroys the FFDP.

Thus, given our link budget estimates and steady-state calculation of the dust
coverage factor, we find that the mount conductance in our initial estimates
based on the ADL report is consistent with the results from both the steady-
state dust fraction calculation and the link budget analysis, and unlikely to
have changed significantly from launch to present day.

###  5.3 Comparisons with Infrared LLR Observations

We can use the derived CCR temperatures for the steady state and eclipse
simulations, and investigate what happens when an infrared wavelength is used
($\lambda=$ 1064 nm). For this calculation, the thermal coefficient of
refractive index for fused silica is adjusted for the wavelength 1064 nm
(Toyoda and Yabe 1983). In comparison to the steady state performance for the
green laser, we find that at 1064 nm, the central intensity in the FFDP is
$\sim$ 60% compared to the case for an isothermal CCR, with a peak-to-peak
phase variation of $\sim$ 2.3 radians. This marks a factor of 10 improvement
in FFDP in full-moon illumination conditions for an infrared laser compared to
the green laser. This agrees with improved results seen by using the infrared
laser for LLR at the Grasse laser station when ranging when the lunar phase is
within $\pm 18^{\circ}$ of the full moon. (Courde et al. 2017). Note that this
only includes the improved central intensity in the FFDP of the CCR, whereas
observations will also depend on other factors such as detector quantum
efficiency, atmospheric throughput and laser power, similar to the link budget
calculation in Section 2.

###  5.4 The Effect of Changing Dust Fraction Coverage on the CCRs

Changing the dust fraction changes the equilibrium temperatures of the CCR, as
well as the resulting gradients under solar illumination. The expected FFDP
for the equilibrium temperature shows that the total intensity inside the
central maximum increases as dust fraction is decreased. A smaller dust
fraction creates a smaller vertical thermal gradient in the CCR, which
improves the return rate of the signal and vice versa. This return rate should
become 100% as the total dust fraction goes to zero. Given a dust fraction,
the qualitative behavior of the eclipse is very similar to the one presented
in Figs. 5 and 8. We find that a dust fraction of 0.5 best matches the eclipse
observations.

We also note that the three reflectors Apollo 11, 14 and 15 show remarkably
similar improvements during the lunar eclipse. These reflectors are placed at
significant distance from each other and were placed over the course of a few
years. The similarity indicates that if the dust obscuration is taken as the
reason for the degrading performance, then this dust deposition must be
uniformly taking place across the surface of the moon. In this case, the dust
transport results from the Apollo 17 LEAM experiment (Berg et al. 1976)
combined with the experimental observations of dust levitation under a plasma
sheath suggests a deposition mechanism (Arnas et al. 2001, Sickafoose et al.
2002, Halekas et al. 2011). The obscuration is likely caused by large
particles, as indicated by the LEAM experiment, deposited due to solar wind
and radiation during lunar sunrises and sunsets creating a plasma sheath,
which transports these dust grains at low altitudes ($<$2 m). Thus, our
results are consistent with the lunar model of dust transport, where the
transport is caused by electrostatic charges on dust particles enabling
transport at low altitudes ($<1$ m) due to the potential difference created at
the time of sunrise.

##  6 Conclusions

Measurements with the APOLLO ranging system indicate a deficit in the return
rate of Apollo lunar retroreflectors by a factor of 16 to 19, with an
additional order-of-magnitude loss when the reflectors are under full moon
illumination. We attribute the degraded reflector performance to the
accumulation of lunar dust on the reflector surface, which partially blocks
lunar ranging laser light and distorts the CCR FFDP by thermal lensing due to
absorbed solar radiation near full moon. This conclusion is supported by three
independent studies:

  1. 1.

A detailed link analysis was undertaken, including a measurement of the APOLLO
transmit laser beam profile, and the one-way throughput of the telescope
optics and the receiver. This study found that the measured lunar return rate
was only 6% of the predicted value, corresponding to a dust fraction of
${\sim}50$%, which in turn results in thermal consequences under solar
illumination.

  2. 2.

Measurements during lunar eclipses reveal that the lunar return rate is highly
dependent on solar illumination. For example, during one lunar eclipse on a
photometric night, we found that the return rate improved rapidly once the
reflectors entered the Earth’s shadow, and subsequently degraded when the
reflectors exit the shadow. When the reflectors are in shadow during a lunar
eclipse, the median return rate is an order of magnitude higher than full-moon
non-eclipse observations.

  3. 3.

We developed a heat transfer model to include conductive and radiative effects
and used the simulated temperature profile of the CCR to generate the
associated FFDP. We explored two scenarios:

    1. (a)

Steady-state study: We ran our simulation until we achieved the equilibrium
state of the CCRs during full moon solar illumination. In the equilibrium
state, the thermal gradient across the CCR degraded the intensity of the
FFDP’s central lobe by a factor of ${\sim}16$ compared to an isothermal CCR
when a 50% dust fraction was used, in agreement with previous assessments of
the full moon deficit Murphy et al. (2010).

    2. (b)

Time-dependent eclipse study: We then ran the equilibrated CCR setup through
an eclipse simulation, roughly matching our observed eclipse conditions. Using
the same dust coverage fraction of 50%, the eclipse response improves over the
full-moon return rate by a factor of 7, which compares well to the temporal
profile observed during the 2014 lunar eclipse.

Taken together, the studies above tell a consistent story: the Apollo 11, 14
and 15 reflector surfaces are partially obscured by dust (${\sim}50$%), which
both reduces their optical throughput by the direct absorption of laser
ranging photons, and through thermally induced distortions of the CCR FFDP due
to the absorption of solar energy near the full moon phase.

## Acknowledgements

We thank Russet J. McMillan for her continued support as APOLLO’s operator
during observation, and handling of ACS functions during ranging. NRC’s
research was supported by an appointment to the NASA Postdoctoral Program at
the Goddard Space Flight Center, administered by Oak Ridge Associated
Universities under contract with NASA. This work is based on access to and
observations with the Apache Point Observatory 3.5-meter telescope, which is
owned and operated by the Astrophysical Research Consortium. This work was
jointly funded by the National Science Foundation (PHY-0602507, PHY-1068879,
PHY-1404491, PHY-1708215) and the National Aeronautics and Space
Administration (NNG04GD48G, NAG81756, NNX12AE96G, NNX15AC51G, 80NSSC18K0482).
JBRB acknowledges the support of the NASA Massachusetts Space Grant
(NNX16AH49H).

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##  Appendix A Thermal Simulation Details

###  A.1 Setting up the Geometry

Defining the geometry of the problem requires us to define a number of thermal
properties for all elements of the 3 dimensional cartesian grid. The CCR are
defined in detail in ADL Inc. report (Arthur D. Little, Inc. 1969) and for the
purposes of the simulation, we consider the properties defined therein to be a
starting point for our simulations. We have a grid of size $51\times 51\times
51$ (102 mm $\times$ 102 mm) over which these properties are defined.

While the code as written can be generalized to an arbitrarily shaped cuboid,
we work here with cubes for simplicity in view factor calculations later.
While we have run the code at different spatial resolutions for testing
purposes, all runs for science purposes have been run at a resolution of 2 mm,
which makes our simulated box size $102\times 102\times 102$ mm. Since the
heat transfer is an explicit finite difference scheme, given the material
properties used and for ease of comparison for different runs, we have fixed
the temporal resolution at 0.005 seconds for all runs. Since the simulation
uses cubic voxels, we have oriented the CCR along the (1,1,1) direction, in
order to create smooth planes for the sides of the CCRs. While this results in
voxelization of the planar surfaces, the effects on heat transfer are minimal.
There are two kinds of numerical effects that might occur as a result of
voxelization of a planar surface of the CCR. Firstly, the amount of radiative
heat transferred between two elements may be slightly inflated due to the
increase in surface area as a result of the voxelization. This effect
decreases with a decrease in element size, theoretically going to zero as the
size of the voxel becomes infinitesimally small. We tested our setup by
creating two parallel slabs oriented along the (1,1,1) direction and
prescribed a heat flux on one side of one of the slabs, which then radiatively
transferred heat to the other slab. Since this is a problem with an analytic
solution, we were able to compare our numerical results with the analytic
solution and found them to show reasonable agreement with each other. The
second effect we need to be careful about is the amount of incident solar flux
illuminating each voxel. We tackled this problem by analytically prescribing
the heat flux for the entire voxel by multiplying the incident solar flux with
the projected area along the (1,1,1) direction.

We first address the assumptions required to simulate a single CCR instead of
an entire retroreflector array. The retroreflector array housing the CCRs on
the moon is machined out of Aluminum 1100. One of the primary goals of the
design of the reflector array was to provide passive thermal control, since it
was understood any thermal gradients would have a significant impact on
reflector performance. Thus, the array as a whole was thermally isolated from
the lunar environment with multiple blankets, reducing external emissivity to
0.01. Additionally, the CCRs are mounted on Teflon rings and the cavities are
recessed into the array to prevent illumination from sunlight unless directly
overhead. These designs work well enough that we can treat an individual CCR
as a standalone retroreflector for thermal purposes, as we will demonstrate
with our simulation. As a result, we have simply replaced the array with an
outer layer of aluminum 1100 which we refer to us as the ”Shell” in this
paper. As we will show, the absence of the remainder of the array has no
impact on the thermal performance of the CCR. Thus, our geometry consists of a
fused silica CCR, two Teflon rings, the upper aluminum retainer ring and the
outer shell. The material properties for these are described in Table 3.

The heat transfer is an explicit finite difference scheme. We first define the
array $K_{i,j,k}$, which is total conductance between adjacent cells. So for a
cell with indices (i, j, k) and a side length $\rm{\Delta}x$, with thermal
conductivity denoted by $\rm{\lambda}_{i,j,k}$, the conductance with respect
to its adjacent cell (i, j, k+1) can be calculated as -

$K_{i,j,k+1/2}=\frac{{\Delta x}^{2}}{\frac{\Delta x}{2k_{i,j,k}}+\frac{\Delta
x}{2k_{i,j,k+1}}+R_{i,j,k+1/2}}$ (2)

Where $R_{i,j,k+1/2}$ is the contact resistance between adjacent cells. This
is zero for adjacent cells with identical thermal conductivity, i.e. in the
bulk of the material. For elements with distinct values of thermal
conductivity, we prescribe the value of contact resistance based on the total
conductance allowed between the respective surfaces. Based on the results of
ADL report, the CCRs were designed to have a prescribed amount of total
conductance between materials and the testing of the CCRs tried to constrain
the total conductance value. For our purposes, we took the values of total
conductance given in the ADL report as starting values. For pairs of material
where no value was given, we prescribed the total conductance as follows. We
calculated the total conductance in two limiting cases. Firstly, when the two
adjacent cells have perfect contact and the total conductance comes purely
from the difference in thermal conductivity of the two cells, we can set the
value of contact resistance to zero. This gives a minimum conductance value.
In the other limit, we assume the two contact surfaces are separated by an
infinitesimal distance and the ”conductance” is a result of only radiative
transfer between the two surfaces. This gives us a temperature dependent total
conductance value, assuming the temperature difference between the two
surfaces is small compared to the absolute temperature. We then calculate this
conductance for the expected temperatures on the lunar surface and get an
upper limit for the total conductance. For the actual simulation runs, we end
up taking the average of these two conductance values as a starting point.
Once we have prescribed the total conductance for a pair of materials, we can
then calculate the conductance per element. Knowing the thermal conductivity
values, we can then prescribe a value of contact resistance for each element.
Thus, prescribing the total conductance between any pair of materials, as
designed and tested in the ADL report, gives us a value for the contact
resistance between any two elements.

###  A.2 Constructing the View Factor matrix

Once we have defined our geometry, the code then constructs a view factor
matrix. We create a matrix of all boundary point elements and compute the view
factors for each pair of surfaces using the analytic expressions from Ehlert
and Smith. Since our geometry does not change as a function of time, this is a
one time calculation performed at the beginning of a run. It can then be saved
and retrieved for a subsequent run. This step of the code takes about 20
minutes to run for the full CCR simulation. We checked our view factor
calculations for parallel and plane geometry by constructing two slabs in a
parallel and perpendicular configuration. We calculate the view factor matrix,
and using the view factor matrix, sum the individual view factors to obtain
the total view factor for the slabs, which can then be computed using the same
expression for a larger slab. We can also check this against simpler
expressions for the special case where the slabs are exactly opposite each
other with no relative lateral displacement. We find that the view factors
computed are consistent when summed with what we would expect for the pair of
larger slabs. We also find the consistency does not depend on the spatial
resolution of the geometry, as would be expected for an analytic expression
for the view factors.

###  A.3 Heat Transfer

Given the conductance between adjacent cells, we can calculate the total heat
influx into a cell $H_{i,j,k}$ from

$\displaystyle\begin{split}H_{i,j,k}=&K_{i-1/2,j,k}\cdot(T_{i-1,j,k}-T_{i,j,k})+K_{i+1/2,j,k}\cdot(T_{i+1,j,k}-T_{i,j,k})\\\
+&K_{i,j-1/2,k}\cdot(T_{i,j-1,k}-T_{i,j,k})+K_{i,j+1/2,k}\cdot(T_{i,j+1,k}-T_{i,j,k})\\\
+&K_{i,j,k-1/2}\cdot(T_{i,j,k-1}-T_{i,j,k})+K_{i,j,k+1/2}\cdot(T_{i,j,k+1}-T_{i,j,k}).\end{split}$
(3)

The change in temperature $\rm{T}$ in time $\rm{\Delta}t$ is given by

$T^{t+1}_{ijk}=T^{t}_{ijk}+H_{ijk}\frac{\Delta t}{{\Delta
x}^{3}\rho_{i,j,k}C_{i,j,k}},$ (4)

where $\rho_{i,j,k}$ is the density and $C_{i,j,k}$ is the Specific Heat
Capacity for the element $(i,j,k)$. For every boundary point, we also track
the temperature at the center of the external surface. In order to check our
heat transfer code, we performed a number of tests. Our general goal was to
run simulations with analytic solutions in order to have something concrete to
compare to. To test the conduction code, we use a single slab and subject it
to an incident flux on one side. We then switch off any radiative effects on
the boundaries, thus reducing this to a one-dimensional heat transfer problem
with a known solution. We can start the simulation run from an arbitrary
temperature and let it run until we have equilibrium. In practice, we started
approximately $50^{\circ}$ away from the steady state temperature value. We
compared the temperatures on the front and back surfaces with the analytically
calculated expected temperatures. We found our simulation agreed with the
analytic solution to well within $0.5\%$ for the coarsest resolution of 1 cm
and to well within $0.1\%$ using the finest resolution of 1 mm. We then tested
our radiative heat transfer code by adding a second slab in parallel behind
the back face of the first slab, with boundary radiative effects switched off.
We had an incident flux on one side of the first slab. Now we can analytically
compute 4 temperatures on the front and back surfaces of the two slabs as
follows. Assuming the temperatures on the front and back of the first slab are
$T_{1}$ and $T_{2}$, and that on the front and back of the second slab are
$T_{3}$ and $T_{4}$, the relevant equations in steady state are Q - σT_1^4 = -
k (T_1 - T_2)
\- k (T_1 - T_2) = σT_2^4 - F_23 σT_3^4
F_23 σT_2^4 - σT_3^4 = - k (T_3 - T_4)
\- k (T_3 - T_4) = σT_4^4, where $k$ is the thermal conductivity of the slab,
$\sigma$ is the Stefan-Boltzmann constant, $Q$ is the incident heat flux per
incident area on the front surface and $F_{23}$ is the view factor between the
back and front surfaces of slabs 1 and 2 respectively. Solving these sets of
equations numerically gives only two real sets of values, only one of which is
physical (positive temperatures). We calculated these values and found good
agreement between the numerically obtained and the solutions to the analytic
equations, with the agreement improving with higher numerical resolution.