Download surface temperature on the moon

ESTIMATION OF THE SURFACE TEMPERATURE OF FLAT AREAS ON THE
MOON
Xiongyao Lia, Yongchun Zhengb,#, Qingxia Lic, Dan Zhangc, Yi Liaoc, Shijie Wanga,
Liang Langc, Shaohong Fua
Email address: [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected]
[email protected] [email protected]
a The State Key Laboratory of Environmental Geochemistry, Institute of Geochemistry,
Chinese Academy of Sciences, Guiyang 550002;
b The National Astronomical Observatories, Chinese Academy of Sciences, Beijing,
100012 Corresponding author: [email protected]
c Department of Electronics and Information Engineering, Huazhong University of
Science &Technology, Wuhan, 430074 China
ABSTRACT
Lunar surface temperature (LST) is affected by the solar irradiance, earthshine and
heat flow for the flat areas on the moon. We present an improved transient temperature
model to calculate temperatures of lunar flat surfaces. The model consists of
one-dimensional thermal diffusion equation and two boundary conditions. The
boundary condition at the lunar surface is significantly improved by exactly deducing
the time-dependent effective solar irradiance (ESI) and earthshine. The error of the ESI
varies from 0 to 3.89 W·m-2 and the theoretical erroneous percentage of this ESI model
is estimated to be less than 0.28% during 100 years from 1950 to 2050.
The simulated surface temperatures suggest consistency with the measured
temperatures from the thermocouples of Apollo 15 and 17 heat flow experiments. From
LST simulated with the improved model, it is found the annual-seasonal variations
present obvious latitude characters, with highest surface temperature occurring in late
October, November and December separately at high (>~72 degree) latitudes, middle
latitudes and low (<~20 degree) latitudes; the lowest surface temperatures occur in
late July. Moreover, the maximum and minimum lunar surface temperatures become
lower with the increasing latitude as well as the differences of the maximum and
minimum temperatures. A 1322.5 w/m2 change of the ESI would lead to 179.4K
change in surface daytime temperature; A 0.12 and 0.02w/m2 change of the earthshine
and heat flow would lead to 0.5K and 0.09K in surface nighttime temperature,
respectively.
Key words: Moon; lunar surface temperature (LST); effective solar irradiance (ESI);
1
heat diffusion equation; model.
2
1. Introduction
Lunar surface temperature (LST) is a quantity of special interest for interpreting
the thermal character of the regolith from the data of the lunar passive microwave
remote sensing and thermal infrared remote sensing. The thermal radiation energy is
usually expressed as the brightness temperature which is directly related to the LST
and effective emissivity about the surface layers of the moon. Hence, the LST is
essential for extracting physical information of the lunar regolith from brightness
temperature. Meanwhile, the LST is regarded as a basic boundary condition for the
thermal evolution model of the lunar interior [Hapke, 1996]. LST could be obtained by
two categories of techniques: one is the experiment related techniques; the other is the
estimation based on physical models.
Direct measurements of the LST have been derived from in-situ Measurement,
estimating by the temperature of landing cabin and the thermophysical properties of
lunar samples, ground-based observations and spacecraft-based observations. The
in-situ measurement (Apollo 15 and 17 sites) were successfully done. The results
deduced from the thermocouple temperatures show the maximum temperature is 384K
with a minimum of 102K at Apollo 17 site, and about 10K higher than the Apollo 15
surface temperatures through most of the night. Lucas et al. inferred the LSTs from
the temperatures of landing cabins of SurveyorⅠ, Ⅲ, Ⅴ, Ⅵ, Ⅶ and the thermophysical
properties of lunar samples collected in Apollo 11, 12 and 17 missions. Lawsow et al
discussed the relationship between the temperature and thermophysical properties,
such as the emissivity, reflectivity, and thermal conductivity via the infrared
1
exploration data obtained in Clementine and Lunar Prospector missions. Nevertheless,
Hagermann et al. analyzed the temperature changing with the heat flow and latitude.
Measurements of LST by ground-based observations mainly utilize the remote sensing
with infrared and microwave, which were concentrated in the lunar equator. The
maximum temperature was related to the center of lunar disk with the minimum to the
rim. In order to recognize the LST distribution, Diviner Lunar Radiometer Experiment
(DLRE) has been executed in NASA’s Lunar Reconnaissance Orbiter mission in 2008.
The DLRE will chart the temperature of the entire lunar surface at approximately 500
meter horizontal scales to identify cold-traps and potential ice deposits.
The experiment related techniques are inadequate. The in-situ measurement is
very expensive, and it can only measure the temperature at a few lunar sites in a short
period. The estimation by the temperatures of landing cabins and the thermophysical
properties of lunar samples could only offer a few data of LST because the raw data was
insufficient. Except the temperatures obtained in Apollo 15 and 17 missions, most of
the data came from the remote sensing, including spacecraft- and ground-based
observations. The spacecraft exploration with higher spatial resolution can detect the
more accurate temperature, but the high cost limits the exploration which need pay a
long time. The low spatial resolution and the influence of Earth atmosphere in
ground-based observations lead to errors which could only reflect the mean
temperature of a large area, and only the lunar nearside could be measured because of
the movement property of the Moon. Therefore, in order to recognize the LST
distribution and variation of the entire Moon, the physical models were developed.
2
Over the years, several models have been developed by different researchers to
model the heat conduction of lunar regolith. Wesselink [1948] first applied the well
known heat conduction equation to an infinitesimal element of volume at the lunar
surface. As the time goes by, the conditions that the heat conduction equation applied
to are more complex. Our model is improved based on Mitchell’s model by accurately
deduced the solar irradiance and earthshine.
In this paper, an improved transient temperature model including the
time-dependent solar irradiance is proposed. Diurnal temperature variation in the
surface layers are determined by considering the balance between the solar radiation,
earthshine, heat flow and the radiation energy from lunar subsurface. One-dimensional
thermal diffusion equation and two boundary conditions are applied to estimate the
temperature. The prominent part of our study is the boundary condition which
incorporates rigorous derivation of the effective solar irradiance and earthshine.
To verify the proposed model in this paper, the surface temperature measured at
Apollo 15 and 17 sites are used to compare with the calculated temperature. The two
most conspicuous and important topographic features near the Apollo 15 heat-flowexperiment site are Hadley Rille and Apennine Front. Apollo 15 landing site can be
approximately regarded as the flat area because the topographic effect of Hadley Rille
is approximately canceled by the effect of the Apennine Front [Langseth et al., 1972].
At Apollo 17 site, the heat flow experiment site is on a local topographic high, perhaps
an intercrater ridge between two wide but shallow depressions north and southeast of
the site. The two probes are implanted near the rim of the shallow northern depression
3
[Langseth et al., 1973]. Accordingly, we model the LST of Apollo 15 and 17 sites with
negligible topography.
2. Improved Transient Temperature Model
We model the LST by considering the basic theory of thermal conduction for the
semi-infinite solid and thermophysical properties of lunar regolith, namely solar albedo,
infrared emissivity, bulk density, thermal conductivity, and heat capacity. The
thermophysical properties change abruptly near the surface, as evidenced by rapid
cooling of the uppermost layer just after sunset followed by slow cooling of the surface
during the night [Vasavada et al., 1999].
A model of the lunar regolith has been constructed which solves the
one-dimensional thermal diffusion equation
 ( x , t ) c ( x, T )
T

T

[ K ( x, T )
]  Qt ( x, t ) ,
t
x
x
(1)
using a finite difference method described by Carslaw and Jaeger [1959]. ρ is the bulk
density as a function of depth; c is the heat capacity as a function of temperature; K is
the thermal conductivity as a function of depth and temperature; Qt is the thermal
radiation generated by translucent media; t is the time.
The sizes and packing of grains likely account for the different modes of
conduction [Ledlow et al., 1992]. The lunar regolith, especially the surface layer, is an
extremely good insulator. The model consists of a 2 cm thick top layer that is highly
insulating and a lower layer that is more conductive, which differ in thermal
conductivity [Vasavada et al., 1999]. We write the thermal conductivity in the form
K  K c [1   (
T 3 ,
) ]
T350
4
(2)
At the top 2cm, K c  9.2 104Wm 1  K 1 ,   1.48 . While at the bottom layer,
K c  9.3 10 3Wm 1  K 1 ,   0.073 .
Heat capacity is a function of only temperature, taken from Jones et al.[1975].
They derived an expression based on lunar Apollo 11 sample measurements.
Following Jones et al. (1992), we represent the heat capacity by
C  0.05277  0.15899  10 2  0.03366  10 4  0.03142  10 7
(3)
Although this equation is based upon Apollo 11 fines, there is essentially no
difference between the specific heat capacity values for the fines and the measured
values for samples of solid materials from the various Apollo sites. The equation
expresses the heat capacity in units of W h kg-1 K-1.
For our application to model calculation we are only concerned with the first few
meters of the subsurface. We assume that the surface material is inhomogeneous down
through the depth to which the diurnal heating wave penetrates. Carrier et al. [1973]
used a hyperbolic function to describe the increase of density with depth. We follow
Carrier and adopt the expression:
  1.92
x  12.2
,
x  18
(4)
where x is the depth under the surface in cm and  is in g/cm3.
To solve the thermal diffusion equation, two boundary conditions are applied.
1) The boundary condition across the lunar surface is the surface heat balance
equation
KS
T
|s   BTS 4  (1  r )[ I (t )  E (t )]  Q ,
x
(5)
where KsT/x denotes the heat conducted into the subsurface; BTs4 is the radiated
5
energy from the surface, where  is the emissivity of lunar regolith, and B is the
Stefan–Boltzmann constant (5.67×10-8 W m-2 K-4); r is the reflectivity; I(t) and E(t)
are the ESI and earthshine on the lunar surface, respectively. Q is the flux originated
from lunar interior.
2) The second boundary condition [Mitchell and pater, 1994] is that at a certain
depth within the lunar regolith where the temperature is merely determined by internal
heat sources; mathematically this means
T
Q
|d 
 1 ,
x
Kd
(6)
where T/x|d and Kd are the temperature gradient and thermal conductivity at the
equilibrium depth; Q is neglected at the thermal equilibrium depth.
The heat diffusion equation (1) can only be solved by numerical methods because
the boundary conditions are too complicated and nonlinear for an analytical solution. A
finite difference formulation is used. The explicit finite difference form of the heat
diffusion equation is expressed as
T ( x  x, t  t ) 
K ( x, T )t
[T ( x  x, t )  2T ( x, t )  T ( x  x, t )]  T ( x, t ) ,
 ( x, t )c( x, T )x 2
(7)
3. Parameters for the Improved Transient Temperature Model
Parameters in the model are of great importance for the calculation of the surface
temperature. To improve the accuracy of the model, the ESI and earthshine are deduced.
Other physical parameters such as the heat flow are selected from the analysis of
previous studies.
6
3.1. Effective Solar Irradiance
Solar irradiance on the lunar surface is the key to understand LST distribution. In
this study, we have constructed a lunar surface ESI real-time model in terms of the
relationship between the solar irradiance, solar constant, solar incidence angle and the
Sun-Moon distance.
The moon in the Solar System rotates on its axe, leading to diurnal changes in Sun
angle and energy input. The ESI on the lunar surface could be represented by (Fig. 1)
Sun
Effective
solar
irradiance
irr Sola
ad r
ian
ce
reflection
i
Lunar surface
Fig.1 Solar irradiance on the lunar surface
I  Esm  cos( i) ,
(8)
where I and Esm are the ESI and total solar irradiance, respectively. The ESI is the
normal part of the solar irradiance, and i is the incidence angle.
Under the assumption that the energy decrement, which is produced by the
absorbing, scattering of other orbs or cosmic dusts, is neglected, the total solar
irradiance absorbed by the surface is therefore [Maxwell, 1998; Owczarek, 1997]
Esm  S0 Rsm- 2 ,
(9)
where, S0 is the solar constant. Rsm is the dimensionless Sun-Moon distance relative to
1AU.
7
Moon
ecliptic
Rem
jem
Rsm
Earth
jsm
Sun
Fig.2. Geometric relationship in the Sun-Moon system
According to the location relationship among the Sun, the Earth and the Moon
(Fig. 2), the Sun-Moon distance could be described by the equation
Rsm  Rem sinjem sinjsm ,
(10)
where Rem is the dimensionless Earth-Moon distance relative to 1AU, and φem is the
geocentric ecliptical latitude. They could be obtained from the lunar orbital
semi-analytic theory ELP2000-82 [Bretagnon and Francon, 1988]. φsm is the
heliocentric ecliptical latitude. It was given by Meeus [1991], in the form
jsm  jem Rem Rse ,
(11)
where Rse is the dimensionless Sun-Earth distance relative to 1AU. It could be deduced
from the planetary orbital theory VSOP87 [Chapront-Touzé and Chapront, 1983].
Hence, the lunar surface total solar irradiance could be expressed as
Esm 
S0 sin 2 j em Rem Rse 
2
Rem
sin 2 j em
8
,
(12)
Measured point
i
Measure point
Rmoon
a
M
Rmoon a
i
O
Lunar equator
K Q
Subsolar point
O
O’
N
b
P
b
Rsm
Subsolar point
Moon
S
Sun
Fig.3 Schematic diagram of solar radiation incidence angle on the lunar surface. O is
the geometric center of the Moon. Plane KOQ is located on the equatorial plane of the
Moon. Plane NO′P is parallel to the equatorial plane of the Moon. K and N are the
projections of M on each plane. Q is the projection of P on the equatorial plane of the
Moon. ∠ KOQ and ∠ NO′P are the difference between the selenographic longitudes of
measured point and subsolar point. ∠ MOK and ∠ POQ are selenographic latitudes of
measured point and subsolar point, respectively.
In Figure 3, the incidence angle of level plane on the lunar surface could be
defined as follows:
i a  b ,
(13)
a  arccoscos jn cos jd cos n  d   sin jn sin jd ,
(14)
where
1/ 2
2

 Rem
sin 2 j em
 Rem Rmoon sin j em cos a  
2
b  arcsin  Rmoon sin a  2
 Rmoon 
  ,(15)
sin
j
R
R
sin
j
R
R





em em
se
em em
se

 

Rmoon is the radius of the Moon. ψn and φn are the selenographic longitude and
selenographic latitude of the measured point, severally; ψd and φd are those of the
subsolar point.
9
According to the analysis above, we can therefore calculate the ESI as
I
S0 sin 2 jem Rem Rse  cos( a  b )
2
Rem
sin 2 jem
,
(16)
where α and β could be solved from equations (14) and (15), respectively; ψd, φd, φem,
Rem and Rse could be obtained from corresponding astronomical algorithms. Because
the geocentric ecliptical latitude of the Moon is quite small （0≤φem≤5.15º）[Duke, 1999;
Heiken et al.,1991], the lunar-surface ESI could be simplified as:
I  lim
jem 0
S0 sin 2 jem Rem Rse  cos( a  b )
R sin jem
2
em
2

S0cos( a  b )
,
Rse2
(17)
Reliable determination of solar constant requires the construction of a composite
record utilizing overlapping data for cross calibration of measurements from different
radiometers, including HF/Nimbus, ACRIM1/SMM, ERBE/ERBS, ACRIM2/UARS,
ACRIM3/ACRIM-Sat, VIRGO/SOHO, and SORCE/TIM [Foukal et al., 2006;
Fröhlich and Lean, 1998; Fröhlich, 2002; Willson and Mordvinov, 2003]. The model
of PMOD developed by Fröhlich showed the solar constant varies between 1361.8 and
1368.2 W·m-2. The average value of the solar constant is about 1366W·m-2 by satellite
observations. The Sun-Earth distance ranges from 0.98AU to 1.02AU [Heiken et al.,
1991]. The variation of lunar-surface ESI caused by the variation of Sun-Earth distance
is larger than 100 W·m-2. The solar radiation incidence angle ranges from 0 to 90 degree.
As a result, the lunar-surface ESI ranges from 0 to 1425.7W·m-2. In the previous
thermal heat transfer models, investigators regarded the ESI as a physical quantity
which is merely related to the solar constant, the Sun-Moon distance and the
elevation angle of the Sun. However, from equation (16), it could be seen that the ESI
10
is inclusive of the information of the longitude and latitude. Hence, the ESI is the
embodiment of the location- dependent and seasonal variation characters. Fig 4
shows the ESI variation at different latitude regions from Apr. 6th, 1971 through Apr.
5th, 1972.
Effective solar irradiance(W/m2)
1500
89.5N
60.5N
30.5N
0.5N
1000
500
0
0
50
100
150
200
Time(day)
250
300
350
400
Fig.4 The ESI variation at 0.5N, 30.5N, 60.5N and 89.5N from Apr. 6th, 1971
through Apr. 5th
3.1.1. Error Analysis
According to equation (8) and (9), the error of the ESI is caused by the solar
constant, Sun-Earth distance and solar incidence angle. By means of the error transfer
theory, systematic error equals to the modulus sum of each variable error multiplying
by its derivative. Therefore, the error is represented by the equation

I
I
I
S 
 R  i ,
S0
Rsm
i
(18)
where S, R and i are the errors of the solar constant, Sun-Earth distance and solar
incidence angle, severally. The variation is greatly related to the solar activity. Some
research indicated that the solar constant shows a tendency of 11-year cycle with
11
strong/weak alternation. Its variation range is from1361.8 to 1369.2W·m-2. We take the
average solar constant (that is S0=1365.5±3.7W·m-2) and its error is
 S  3.7W / m 2 .
(19)
On the basis of VSOP87 theory, we present the error of the Sun-Earth distance
 R  9.13Rsm  106 cy 1 .
(20)
The error of the solar incidence angle mainly comes from a of equation (13) , because
the Sun-Moon distance is larger than the lunar radius. It leads b of equation (15) to be
very small. The error of the solar incidence angle is therefore
 i  10 9 cy 1 .
(21)
When equation (19), (20) and (21)are substituted into equation (18), it is


3.7 cos i 18.26S0 cos i
S sin i

 10 6  0 2  109  N .
2
2
Rsm
Rsm
Rsm


(22)
where N is the number of years relative to 12 o’clock on January 1, 2000. According
to the geometric relationship among the Sun, the Earth and the Moon, Rsm reaches the
minimum value of 0.980466AU when the Earth is located at perihelion and the Moon is
on the line between the Sun and the Earth. The solar radiation incidence angle ranges
from 0 to 90 degree. Hence, the error of the ESI varies from 0 to 3.89 W·m-2 and the
theoretical erroneous percentage of this model is estimated to be less than 0.28% during
100 years from 1950 to 2050. These indicate that the model can accurately reflect the
variation of ESI on the lunar surface.
3.2. Earthshine
The earthshine includes the emitted (infrared) radiation of the Earth and the
reflected solar radiation by the Earth (Fig.4). Locked in synchronous rotation, the
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Moon always hides its farside away from Earth and faces Earth only with its nearside.
The earthshine just influences the LST at the lunar nearside and can be described as
terrestrial albedo and emissivity. Hence, the energy conservation of a semi-sphere
could be written as:
2Rem  E  ( AS0  Te )  2Re
2
4
2
(23)
where, A is the terrestrial albedo, and Re is the radius of the Earth. Equation (23) can be
simplified to the following:
E  ( AS 0  Te ) 
4
Re
2
Rem
(24)
2
earth
emission
reflection
S
n
tio
dia
a
r
r
ola
Lunar thermal
radiation
earthshine
Sun
So
lar
rad
iati
on
Lunar surface
Heat flow
Fig.4 The solar radiation, heat flow, earthshine, and lunar thermal radiation influence
the LST of a horizontal plane on the Moon.
As the Sun, the Earth and the Moon are located on different sites, the value of
reflected radiation ranges from 0 to 0.11 W·m-2. When the Earth is between the Sun
and the Moon, and the Moon is just in the shadow of the Earth, a given site on the
nearside of the Moon is blocked from the solar radiation and receives only terrestrial
infrared radiation, that is, the reflected radiation is nearly zero. Assumed the average
13
temperature of the earth is 287K. Based on the data from the Big Bear Solar
Observatory since December 1998, Goode et al. [2001] suggested the average albedo
is 0.297±0.005. With a average radius 6371 km of the Earth, a average Earth-Moon
distance of 384402 km, and a solar constant 1366W·m-2, thereby the infrared
radiation received by the Moon is 0.099 W·m-2. Under most circumstances, the
earthshine that a given site on the nearside of the Moon receives from the Earth is
about 0.099~0.201 W·m-2. In this paper, we choose 0.12 as the value of the
earthshine.
3.3. Heat Flow
The lunar heat flow is important that it provides a basic data for inferring thermal
state of the lunar interior and is directly related to the abundance of radiogenic
elements. Measurements made at Apollo 15 and 17 sites show the heat flow at these
two sites are 0.021±0.003W·m-2 and 0.014±0.002 W·m-2, respectively [Langseth et al.,
1976]. A global variation range was estimated to be 0.02W·m-2 to 0.04W·m-2 based on
the measurements at Apollo 15 and 17 sites and Earth-based observations of
microwave emission [Heiken et al., 1991]. Because the thermal skin depth is small,
the solar radiation has a little impact on lunar subsurface temperature. The impact of
the heat flux is comparatively elevated, especially during the night.
3.4. Thermophysical Properties of Lunar Regolith
Wildey [1977] showed that the reflectivity of the lunar surface layer is about
0.090~0.228 with an average of 0.125. It was estimated to be about 0.090~0.189 at the
visible and infrared bands [Zou et al. 2004]. Furthermore, Racca [1995] obtained the
reflectivity of the lunar surface by fitting the earth-based observation data, which is
14
equal to 0.127. According to the data published by Wildey [1977], the values of
reflectivity at the Apollo 11, 12, 15 and 17 landing sites have been averaged among
adjacent regions. They are 0.102, 0.102, 0.116, and 0.120, respectively. The maximal
change is smaller than 2.3K in the calculation of LST, when the reflectivity ranges
from 0.102 to 0.127 or from 0.127 to 0.228. To simplify the calculations it is necessary
to assume a constant value for the surface reflectivity in the given range, and we use the
reflectivity published by Racca [1995] to calculate the LST in this study.
In the thermal unsteady state, the emissivity isn’t equal to the absorptivity. Li, X et
al. [2000] show the material has a stronger emission in thermal unsteady state than in
thermal steady state. Therefore, the emissivity of lunar surface layer couldn’t be
deduced simply from the reflectivity. Hale et al showed that the characteristics of
thermal emission on the lunar surface are approximate to those of a black-body. Its
thermal emissivity is about 0.90~1.00 [Hale and Hapke, 2002; Salisbury et al., 1997;
Keihm and Langseth, 1973]. To simplify the simulation, it’s also necessary to assume a
constant value for the surface emissivity in the given range. The values have been
averaged and a constant value ε=0.94 has been considered in the simulation. With those
mean values of I(t)=1366W·m-2 for lunar daytime and I(t)=0W·m-2 for lunar nighttime,
E(t)=0.12W·m-2, and Q=0.02W·m-2, the LST could be computed by equation (1).
4. Simulation results and Comparison with Experiments
To verify the proposed model, the simulated surface temperatures are compared
with the measured temperatures at Apollo 15 and 17 sites in this section.
Parameters for simulation
15
The constant parameters used in simulation are shown in Table 1.
Table 1. Constant Parameters in simulation
Parameters
r(reflectivity)
Value
0.127
(emissivity)
0.94
S0(solar constant)
1366
E(t)(earthshine)
0.12
Q(Apollo 15 heat flow)
0.02
Q(Apollo 17 heat flow)
0.014
Experiment data for comparison
Both Apollo 15 and 17 heat flux experiments used the same basic design. Two
probes were implemented, and four thermocouples were placed on the probe cables
which lie on or just above the lunar surface [Langseth et al, 1972]. Langseth et al.
[1973] show the surface temperatures were monitored by thermocouples in cables
placed several centimeters above the lunar surface at the Apollo 15 and 17 sites. These
thermocouples are in radiative balance with the lunar surface, the solar radiation and
space, governed by the flux balance equation. With solving the flux balance equation,
at the Apollo 15 site, the maximum temperature is 374K with a minimum of 92K.
Temperatures at the Apollo 17 site are about 10K higher at night [Langseth and Keihm,
1977].
16
Comparison between simulation and experiment
With the above parameters, the lunar surface temperatures at the Apollo 15 and 17
landing sites are calculated with a numerical method. The results are shown in Fig.5.
measured data
Racca's model
our model
350
Temperature(K)
300
250
200
150
100
0
5
10
15
Time(day)
20
25
30
(a) Apollo 15 landing site
measured data
Racca's model
our model
350
Temperature(K)
300
250
200
150
100
0
5
10
15
Time(day)
20
25
30
(b) Apollo 17 landing site
Fig.5 Comparison of the LST within experimental observation (Keihm and Langseth,
17
1973), Racca’s (1995) simulation and our simulation at Apollo 15 and 17 sites during a
lunation
Fig.5 shows the variation of LST from the simulation is consistent with that from
measurements during the day and night. The variation of the temperature is rapid
during daytime, mainly determined by the solar irradiance because the lunar surface is
lack of atmosphere. At night, the variation of temperature is smooth, and mainly
controlled by the earthshine and heat flow. The simulation gives surface temperature
with a maximum of 372.4K and a minimum of 93.1K which are close to the measured
maximum of 374K and minimum of 92K at Apollo 15 site. The simulation also gives
surface temperature with a maximum of 376K and a minimum of 102.5K at Apollo 17
site, where the measured maximum surface temperature is 3846K and minimum
surface temperature is 1022K. Furthermore, the simulation represents the temperature
is increasing a little slower after dawn and decreasing a little slower after sunset than
the measured temperature values at Apollo 15 and 17 sites. Compared with Racca’s
steady state temperature model, our model is commendably improved the temperature
at night. The nighttime temperatures should gradually decline according to the
thermal conduction theory. However, the nighttime temperatures deduced by the
Racca’s model keep steady.
Calculated surface temperatures from our model in fig 6, the surface
temperatures vary with the time at different latitudes during an earth year. The
annual-seasonal variation can be observed, and the highest surface temperatures occur
in late October, November and December separately at high (>~72 degree) latitudes,
18
middle latitudes and low (<~20 degree) latitudes; the lowest surface temperatures
occur in late July. It is also showed that the surface temperatures have obvious
seasonal variation at high latitudes. Moreover, the LSTs decrease with the increase of
the latitude.
400
89.5N
60.5N
30.5N
0.5N
350
Temperature(K)
300
250
200
150
100
50
0
50
100
150
200
Time(day)
250
300
350
400
Fig 6 The surface temperature variation at 0.5。N, 30.5。N, 60.5N and 89.5。N during
an earth year
400
350
Temperature(K)
300
250
nearside maximum temperature
nearside minimum temperature
farside maximum temperature
farside minimum temperature
the difference of nearside max
and min temperature
200
150
100
50
-100
-80
-60
-40
-20
0
20
Latitude(dgree)
19
40
60
80
100
Fig.7 The surface maximum and minimum temperature variation with the
latitude
Owing to lack of atmosphere, the lunar surface temperature is mainly determined
by the ESI that is closely related to the latitude and solar incidence angle. The
maximum and minimum temperatures become lower as the latitude increases in the
southern hemisphere and northern hemisphere. The variation trend is consistent with
the ESI. The solid line and “*” line show the maximum and minimum temperature at
the lunar nearside and farside, separately. The differences of the maximum and
minimum temperatures display the same variation. Because the lunar farside is not
affected by the earthshine, the night temperature is about 0.5K lower than the lunar
nearside. However, in the daytime the maximum temperatures are basically in
superposition due to the diurnal temperature determined by the solar irradiation.
According to the improved model, the contributions to the change of LST from
the ESI, earthshine and heat flow will be evaluated. Near the equator a 1322.5 w/m2
ESI change from 1322.5 to 0 w/m2 would lead to 179.4K change in lunar surface
daytime temperature; A 0.12 w/m2earthshine change from 0.12 to 0 w/m2would lead
to an about 0.5K change in lunar surface nighttime; A 0.02 w/m2 heat flow change
from 0.02 to 0 w/m2 would lead to an about 0.09K change in lunar surface nighttime.
In addition, the cooling rate is determined by the thermal diffusivity, k/c. The
simulated temperature cooling curve shows a little slower than the measured
temperature cooling curve, which is possibly caused by the thermal diffusivity.
5. Conclusion
Lunar surface temperatures have complicated relationship with the ESI,
20
earthshine, heat flow, and topography. For the flat areas on the Moon, the influence of
topography can be ignored. To simulate the variation of the LST, an improved transient
temperature model which consists of one-dimensional thermal diffusion equation and
two boundary conditions is proposed in this paper. The boundary condition at the lunar
surface was significantly improved by exactly deducing the time-dependent solar
radiation, earthshine, heat flow, and other thermophysical properties. The error of the
ESI varies from 0 to 3.89 W·m-2 and the theoretical erroneous percentage of this ESI
model is estimated to be less than 0.28% during 100 years from 1950 to 2050.The
contributions to the change of LST from the ESI, earthshine, and heat flow are
evaluated. A 1322.5 w/m2 change of the ESI would lead to 179.4K change in surface
daytime temperature; A 0.12 and 0.02w/m2 change of the earthshine and heat flow
would lead to 0.5K and 0.09K in surface nighttime temperature, respectively.
The time variation of surface temperature is simulated with the improved transient
model with typical values of the thermophysical parameters. The simulated surface
temperature is close to measured values at Apollo 15 and 17 sites. The simulated
surface temperature can reflect the change of LST during the lunar day and night. Once
the accurate thermophysical parameters on fixed location of the moon are obtained,
the LST would be probably estimated by this model.
Although the simulated surface temperatures approach the measured ones, there
are still small deviations possibly caused by the ignorance of topography operation in
simulation, the selection of thermophysical parameters, the errors of measured data, etc.
During lunar day, the surface temperature is determined by the solar radiation. It is
21
obvious that the topography of Apollo 15 and 17 sites is not ideally flat and hence the
accuracy of the model is supposed to be affected. At night, the surface temperature is
determined by the bulk thermal inertia, (kc)1/2, of the surface layers ( k,  and c are the
thermal conductivity, bulk density and heat capacity, respectively) which are
accompanied with the influence of inner sources (heat flow). The thermal inertia
describes the resistance of a medium to temperature change. The bigger the thermal
inertia is, the smaller the temperature difference between day and night is. Hence, the
simulated surface temperatures at night are influenced by the choice of the
thermophysical parameters values.
On the moon, most surface area can not be considered as flat. Thus, the
topography is suggested to be considered as a factor in the computation of the
variation of the surface temperature.
22
Acknowledgements:
This work was supported by the National High Technology Research and
Development Program of China (863 Program) (Grant No. 2010AA122200), the
National Natural Science Foundation of China (grant No.40803019, No. 40873055,
and No.40673053), and China’s
Lunar Exploration Program
TY3Q20110029).
23
(Grant
No.
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