**Radiative Non-Equilibrium at the Lunar Surface**

Brendan Hermalyn

2006 Summer Institute on Atmospheric, Biospheric, Hydrospheric,

and Solar and Space Plasma Sciences

NASA/Goddard Space Flight Center, Greenbelt, MD

Abstract

The study of Earth's moon is essential to our future, both for our
eventual return, and for our understanding of our universe. This project
examines the complex relationship between solar radiation and lunar
surface temperature by analyzing data from the Apollo 15 and 17 thermal
experiments and fitting our mathematical models to the data. The project
culminated in the deployment of a web-based application that can
determine the approximate lunar surface temperature for any latitude and
longitude and time in the next hundred years.

Final report

The first part of the project was to conduct analysis on the lunar
surface temperature data. This data was recorded during an experiment
conducted on the Apollo 15 and 17 missions called, simply "heat flow
experiment." The goal of the experiment was to determine the thermal
properties of the top crust of lunar regolith (intended to be
approximately 3 meters by design of experiment, but due to drilling
difficulties, includes only a bit over a meter). The actual surface
temperature readings were not the original intent of the experiment;
however they are of much interest in this project. The thermal
properties of the lunar regolith are also essential, as will be
discussed below.

Both experiments used the same basic design (please see respective
preliminary science reports for more detail.) Two probes were
implemented, each of which used gradient and ring bridges on the probe
itself to measure the heat flow. Four heaters were used in order to
stimulate heat flow in the regolith. In addition, four thermocouples
were placed on the probe cables which lie on or just above the lunar
surface. Each probe has five data files, named a15p1.file_1, and so on;
files 1, 2, 4, and 5 are records of the upper and lower bridges, both
gradient and ring. The file 3 records contain the records of the four
thermocouples, and thus is of the most interest to this project.

Of the sixteen total thermocouples on the Apollo 15 and 17 experiments,
only a few thermocouples yield useful surface temperature readings.
These are, in order of usability, thermocouple 2 from a15p2.file_3,
thermocouple 2 from a15p1.file_3, thermocouple 2 from a17p2.file_3, and
thermocouple 2 from a17p21.file_3. The data analysis in this project was
conducted on the data retrieved by TC2 a15p2.file_3.
![](media/image1.png){width="5.286111111111111in"
height="5.752083333333333in"}

Figure 1. Lunar temperature from Apollo 15 with surface in red, and
sub-surface temperature in green. The earth eclipses are evident in the
red graph.

Because of the regolith's thermal conductivity, the variation between
lunar day and night is almost indistinguishable anywhere over 50 cm
underneath the surface (the Apollo 17 report claims a depth of 100 cm to
remove diurnal variation, but the files do not show the variation at
about 50cm.); even sensors closer to the surface did not read enough
variation to be directly usable. Unfortunately, there was a large amount
of noise in most of the readings; there is a possible correlation
(especially in the Apollo 17 readings) with surface radioactivity due to
thorium abundances. Please see the Apollo 17 Preliminary Science Report,
9-17 for more information.

The a15p2.file_3 file required slight manipulation in order to edit out
obviously out of range data points (including several sub-zero Kelvin
data points). IDL was employed to graph the surface temperature
mutations with a program called plot_data.pro. This program was refined
and heavily modified into several different versions that conducted
various data analysis upon the file. The data was smoothed with the
"smooth" IDL function in plot_smooth.pro, then in a different manner
with a moving average procedure in mvg_avg_fix.pro in order to remove
noise. Care had to be taken to avoid removing the marked downward spikes
that signified lunar eclipses through these smoothing procedures.
Another one of the analyses was to plot one lunar day over another to
observe the differences in length and temperature. This required the
days to be plotted peak-to-peak (completed by utilizing a maximum
function) and resampling and interpolating the data at regular
intervals. This was all accomplished in program plot_any.pro.

An interesting component of the set of data was a series of regular
spikes that occurred during each transition from lunar day to lunar
night. ![](media/image2.png){width="6.153472222222222in"
height="6.401388888888889in"}Figure 2. Examples of lunar surface
temperature with earth eclipse just after the noon. Regular spikes in
temperature just before the sunset are evident.

A possible explanation for these include geographical effects that place
the probe in a shadow; however, their regularity and presence over the
course of the years of collected data tend to cast doubt upon that
interpretation. Although it could easily be explained as equipment
fault, the anomalies should be investigated further. A short list of
exact days and times for the first few anomalies was compiled as an
excel chart under the title Lunar Anomaly Dates.xls.

Once the data was analyzed corrected, the construction of a model
commenced. This section of the project used a previously written Fortran
program called "moonangsHM1.f" that was built upon a JPL ephemeris
program. It calculates the exact spatial location of the moon/sun/earth
system, and then can calculate the relationship between each. The
program was modified to compute the radiation impingent upon any point
upon the lunar surface.

The original output of the program was a large table of the radiation,
zenith angles, and distances computed from dates starting from 1900 up
to and including the 23^rd^ century. The first step here was to modify
the output so that the amount and dates calculated was of a much more
useful size. The moon was then assumed to be a black body (i.e.
radiative equilibrium) and the surface temperature on the surface was
calculated from the radiation (this is the combined radiation from the
sun, the earth, and the reflection from the earth), and the position of
the system using the Stefan-Boltzmann law as $j \ast \text{es}T^{4}$,
where j\* is the irradiance in units of power density, T is absolute
temperature, ε is emissivity of object, and σ is the Stefan-Boltzmann
constant (taken to be approximately 5.670400(40)×10^−8^
[W](http://en.wikipedia.org/wiki/Watt)·m^-2^·K^-4^.) More useful to us
is the form F=F~0~\*cos(solar zenith angle) where F~0~ depends on
Sun-Moon distance and is calculated by the Fortran program.

This computed time series of temperatures was compared with the
corresponding smoothed data from the Apollo experiments. The computed
data correlated very well with the mutations of lunar day and night, but
was not accurate with respect to temperature. (See Figure Below;
predicted surface temperature is in red; actual surface temperature
readings are in green) ![](media/image3.png){width="5.7625in"
height="5.8069444444444445in"}

Figure 3. Observed (red) versus radiative equilibrium temperatures
demonstrating the lunar surface is in radiative non- equilibrium state.

The black body assumption resulted in lunar nights that were
approximately 60K colder than the real data, and lunar days that were
20K hotter. This, as the title of the project suggests, is because the
moon does not function as a black body, but instead employs a more
complex model.

During lunar day, the lunar regolith absorbs the radiation from the sun
and transports it inward and is stored in a layer approximately 50cm
thick. As the moon passes into night, the radiation from the sun quickly
approaches zero (there is still a bit of radiation from the earth) and,
in contrast with a precipitous drop in temperature if it was a simple
black body, the regolith then proceeds to transport the stored heat back
onto the surface, thus warming it up significantly over the black body
approximation. To make this procedure even harder to quantize, the
thermal conductivity of the regolith is both temperature and depth
dependent; it thus reacts differently in the lunar day as it does in
lunar night and at different depths. The heat flow from the lunar core
must also be taken into account.

At this point, several approximations were used, as our knowledge of
inner workings of the moon is hazy at best. Because the drills of the
Apollo missions jammed after approximately one meter, we do not have any
concrete estimates of the thermal conductivity of the lower, more
densely packed lunar regolith; as we know from the top layer, the
thermal properties of the regolith is very temperature dependant. The
amount of heat coming from the center of the moon is also cast into
question. Although a numerical answer was fairly easy to determine with
the experimental setup, the actual mechanism of what creates that heat
is unknown.

All of these different variables result in a differential equation of
the form: \--$\rho C(T)\frac{¶T}{¶t} = \frac{¶}{¶x}(K(T)\frac{¶T}{¶x})$
\-- where ρ is the density, C is the thermal capacity, K is the
conductivity, T is the temperature t is time, and x is the depth. With
some manipulation, one gets the equation
$\frac{1}{k}\frac{\text{¶q}}{¶t} = \frac{¶^{2}q}{¶x^{2}}$ where
$q(T) = \int_{0}^{T}{K(T)\text{dT}}$ and
$k(T) = \frac{K(T)}{\rho C(T)}$.

This is solved using the Crank-Nicholson method in program
Crank_Nicholson1.f

The actual deployment of the web application was also something of a
challenge. Due to the complex computation required to calculate the many
different steps in the process, runtime was a factor that was high on
the priority list; therefore, it was decided to use a further modified
version of the Fortran program as a backend for speed as well as ease of
implementation. The front end of the web application was attempted in a
few different languages, including Perl, Java, and CGI before PHP was
selected for speed. As none of the mathematics happens on the front end,
computational strength was not a requirement.

The implementation of the web application is fairly straightforward: the
user enters in the desired latitude, longitude, and time into a form.
The form is then parsed by the PHP and the contents written to a
temporary file. The PHP invokes the Fortran program ephem_rad.f, which
reads the variables from the temporary file and computes the necessary
data. The program is set up to compute approximately three lunar days
worth of ephemeris and radiation data; this is because the
Crank-Nicholson method requires a sufficient time series of data to make
an accurate computation. The original program was compiled to use five
lunar days, but by trimming that down to three lunar days, runtime was
decreased by a measurable amount with a loss of accuracy of about .08
Kelvin, which should be well within acceptable limits for this
application. The Fortran program writes a file of radiation magnitudes
and outputs the total radiation, radiation due to the sun, radiation due
to the earth, solar and terrestrial zenith angles, solar and terrestrial
distances, and approximate temperature due to the Stefan-Boltzmann law
to the screen. The PHP script captures the Fortran's output and parses
it into a list. The last entry can then be displayed on the web browser
window (the Fortran program generates the time series from three lunar
days before the entered time, and terminates at the entered time.). The
Crank-Nicholson Fortran program is then implemented, The Crank-Nicholson
program interpolates the data built off the three lunar days prior to
the user entered date so that there are hourly samples and then solves
the tridiagonal linear matrix. Finally, the program generates another
time series of temperatures and outputs that to the screen. The PHP
captures this into another array and prints the last line to the screen.

This model is not exhaustive, however, as it makes many assumptions. The
most glaring approximation is that of the lunar surface itself; this
model assumes that it is smooth like a billiard ball. This is, of
course, not the case, as the lunar surface is pot marked with thousands
of impact craters. The surface temperature in the shadow of a crater
would be much lower during lunar day then a point just outside of the
crater in full sunlight. As the moon rotates, geographical
characteristics (i.e. mountains, cliffs, etc) change position and can
plunge a point on the surface into shadow even during the height of
lunar day. As another result of the impact craters, the albedo is far
from constant, since the ejecta, undisturbed regolith, and melted
regolith have different colors and thus different emmisivities. This
could cause local differences of several degrees Kelvin based on a 5%
difference in albedo. Also, as stated above, we know very little about
the inner workings of the moon, and much of the heat transfer constants
used are simply educated guesses. This is one of the many reasons we
need to return to the moon. One possible explanation of the poor data
was that these heat flow experiments were considered as trial runs for
the "real" experiments that were scheduled for the Apollo 18 mission and
beyond. During the design and implementation of these experiments, the
scientists did not assume that this would actually be the best data-if
not the only data- available almost 40 years later.

One of the main reasons to look at the moon from the radiation
standpoint is to get a glimpse of solar variations in the past. Although
there has been much research done on glacial ice samples, the earth's
ecosystem is simply too complex to clearly infer the solar radiation
variations over the past few millennia. The lunar surface, on the other
hand, is a much simpler model, without atmosphere, no running water, and
what is assumed to be an inactive core. By examining the radiation
profile of the moon, we can learn more about our own system on earth.

References:

Colozza, A., Analysis of Lunar Regolith Thermal Energy Storage, *NASA
Contractor Report 189073*, 1991.

Lowman, P. Jr. Field Trip To The Moon, *The Science Teacher*/ NASA.
April 2004.

Langseth, M. G. Jr., S. P. Clark, J. L. Chute., Jr., S. J. Keihm, and A.
E. Wechsler, Heat-Flow Experiment, *Apollo 15 Preliminary Science
Report*, NASA, 1972.

Langseth, M. G. Jr., S. J. Keihm, and J. L. Chute., Jr., Heat Flow
Experiment, *Apollo 17 Preliminary Science Report*, NASA, 1973.

Ozisik, M., Heat Conduction: Second Edition. *John Wiley & Sons*, New
York. 1993.