Lunar surface temperature (LST) is a quantity of special interest for interpreting the thermal character of the regolith. It 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 improved lunar surface boundary condition allows one to precisely determine the effective solar irradiance (ESI) and earthshine. The simulated surface temperatures suggest consistency with the measured temperatures from the thermocouples of Apollo 15 and 17 heat flow experiments. From the LST simulated with the improved model, it is found the annual-seasonal variations present obvious latitude characters, with the highest surface temperature occurring in late October, November, and December separately at high (>∼72°) latitudes, middle latitudes, and low (<∼20°) latitudes, respectively; the lowest surface temperatures occur in late July. Furthermore, the discrepancies between the maximum and minimum temperatures decrease as latitude increases, as do the maximum and minimum lunar surface temperatures. The surface daytime temperature would change by 179.4 K with a 1322.5 W/m2 change in the ESI. A 0.12 W/m2 and 0.02 W/m2 change of the earthshine and heat flow would lead to 0.5 and 0.09 K in surface nighttime temperature, respectively.

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 of the surface layers of the Moon. Because of this, the LST is necessary to derive physical data from brightness temperature about the lunar regolith. Meanwhile, the LST is regarded as a basic boundary condition for the thermal evolution model of the lunar interior.1 LST could be obtained by two categories of techniques: one is ground and spacecraft measurements, and the other is the estimation based on physical models.

Direct measurements of the LST have been derived from in situ measurements, estimated by the temperature of landing cabin and the thermophysical properties of lunar samples, ground-based observations, and spacecraft-based observations. The in situ measurements of the Apollo 15 and 17 sites were completed satisfactorily. The results deduced from the thermocouple temperatures show that the maximum temperature is 384 K with a minimum of 102 K at the Apollo 17 site, and throughout the majority of the night, surface temperatures were roughly 10 K higher than those at the Apollo 15 site. Lucas et al.2 inferred LSTs from the temperatures of landing cabins of Surveyor I, 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 exploration data obtained in Clementine and Lunar Prospector missions. Nevertheless, Hagermann et al.3 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 lunar disk’s center corresponded to the highest temperature, while the rim had the lowest. In order to recognize the LST distribution, Diviner Lunar Radiometer Experiment (DLRE) has been executed in NASA’s Lunar Reconnaissance Orbiter mission in 2008. To locate possible ice deposits and cold spots, the DLRE will map the temperature of the whole lunar surface at a distance of roughly 500 m on a horizontal scale.

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 were 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 operate 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, physical models were developed.

Researchers have created a number of models over time to simulate the heat conduction of lunar regolith. Wesselink4 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 is applied to are becoming more complex. Our model is improved based on Mitchell’s model5 by accurately deducing the solar irradiance and earthshine.

This work proposes an enhanced transient temperature model that takes time-dependent solar irradiance into account. The balance between solar radiation, earthshine, heat transport, and radiation energy from the lunar subsurface is used to calculate the variations in temperature during the day in the surface layers. 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 a rigorous derivation of the effective solar irradiance and earthshine.

To verify the proposed model in this paper, the surface temperatures of the Apollo 15 and 17 sites are utilized to compare with the computed temperature. Hadley Rille and the Apennine Front are the two most notable and significant topographic features close to the Apollo 15 heat-flow experiment site. Because the Apennine Front’s effect roughly cancels out Hadley Rille’s topographic effect, the Apollo 15 landing location can be roughly viewed as a flat area.6 At the Apollo 17 site, situated on a local topographic high, maybe an intercrater ridge between two broad yet shallow depressions to the north and southeast of the site is the heat flow experiment site. In close proximity to the edge of the narrow northern depression, two probes are inserted.7 Accordingly, we model the LST of the Apollo 15 and 17 sites with negligible topography.

We model the LST by considering the basic theory of thermal conduction for the lunar regolith’s semi-infinite solid and thermophysical characteristics, including bulk density, thermal conductivity, solar albedo, and infrared emissivity.8 The uppermost layer cools quickly after sunset, while the surface cools more slowly over the night, indicating that the thermophysical parameters change dramatically close to the surface.9 

A model of the lunar regolith has been constructed, which solves the one-dimensional thermal diffusion equation
ρ(x,t)c(x,T)Tt=xK(x,T)Tx+Qt(x,t)
(1)

using a finite difference method described by Carslaw and Jaeger.10 Here, K is the thermal conductivity as a function of depth and temperature; ρ is the bulk density as a function of depth; c is the heat capacity as a function of temperature; Qt is the thermal radiation generated by translucent media; and t is the time.

The sizes and packing of grains likely account for different modes of conduction. The lunar regolith, especially the surface layer, is an extremely good insulator. The model is composed of two layers with different thermal conductivities: a lower, more conductive layer and a top, highly insulating layer that is 2 cm thick.9 The thermal conductivity is expressed in the form
K=Kc1+χTT3503.
(2)

At the top 2 cm, Kc = 9.2 ⋅ 10−4 Wm−1 K−1 and χ = 1.48, while at the bottom layer, Kc = 9.3 ⋅ 10−3 Wm−1 K−1 and χ = 0.073.

Based solely on temperature, heat capacity is determined.11 They used measurements from the lunar Apollo 11 sample to derive an expression. Following Jones et al.,11 we represent the heat capacity by the following equation:
C=0.05277+0.15899×1020.03366×104+0.03142×107.
(3)

There is basically 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, even though this equation is based upon Apollo 11 fines. 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.12 used a hyperbolic function to describe the increase in density with depth. We follow Carrier and adopt the following expression:
ρ=1.92x+12.2x+18,
(4)
where x is the depth under the surface in cm and ρ 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,
    KSTx|s=εσBTS4(1r)[I(t)+E(t)]+Q,
    (5)
    where KsT/∂x denotes the heat conducted into the subsurface; εσBTs4 is the radiated energy from the surface, in which ε is the emissivity of lunar regolith and σB 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 effective solar irradiance (ESI) and earthshine on the lunar surface, respectively; and Q is the flux originated from lunar interior.
  2. The second boundary condition13 is that at a certain depth within the lunar regolith where the temperature is merely determined by internal heat sources, mathematically,
    Tx|d=QKd1,
    (6)
    where T/∂x|d and Kd are the temperature gradient and thermal conductivity at the equilibrium depth and Q is neglected at the thermal equilibrium depth.
    The heat diffusion Eq. (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+Δx,t+Δt)=K(x,T)Δtρ(x,t)c(x,T)Δx2T(x+Δx,t)2T(x,t)+T(xΔx,t)+T(x,t).
    (7)

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.

The solar irradiation on the lunar surface is crucial for determining the LST distribution. We have built a real-time lunar surface ESI model in this study, taking into account the correlations between sun irradiance, solar constant, solar incidence angle, and Sun–Moon distance.

Because of the Moon’s axial rotation within the Solar System, the Sun’s angle and energy intake vary during the day. The ESI on the lunar surface could be represented as shown in Fig. 1 and can be expressed as
I=Esmcos(i),
(8)
where ESI and total solar irradiance, respectively, are represented by I and Esm. i is the incidence angle, while the ESI is the normal component of solar irradiance.
Under the assumption that the energy decrement, which is produced by the absorption and scattering of other orbs or cosmic dusts, is neglected, the total solar irradiance absorbed by the surface is, therefore,14,15
Esm=S0Rsm2,
(9)

where S0 is the solar constant in this case. The dimensionless Sun–Moon distance in relation to 1 AU is called Rsm.

Based on the position relationship between the Earth, Moon, and Sun (Fig. 2), the following equation might be used to explain the Sun–Moon distance:
Rsm=Remsinφem/sinφsm,
(10)
where φem is the geocentric ecliptic latitude and Rem is the dimensionless Earth–Moon distance in relation to 1 AU. These could be acquired by the semi-analytic theory of lunar orbits ELP2000-82.16 The ecliptic latitude in heliocentric space is φsm. Meeus17 provided it by the following equation:
φsm=φemRem/Rse,
(11)
where Rse, with relation to 1 AU, is the dimensionless Sun–Earth distance. One could infer that from the VSOP87 theory of planetary orbits.18 Hence, the lunar surface total solar irradiance could be expressed as
Esm=S0sin2φemRem/RseRem2sin2φem,
(12)

where O represents the Moon’s geometric center. The Moon’s equatorial plane is home to aircraft KOQ. The equatorial plane of the Moon and plane NO’P are parallel. M’s projections on each plane are K and N. On the lunar equatorial plane, Q is the projection of P. The difference between the measured point and subsolar point’s selenographic longitudes is represented by ∠KOQ and ∠NO’P. The selenographic latitudes of the measured point and subsolar point are denoted by ∠MOK and ∠POQ, respectively.

In Fig. 3, on the lunar surface, the incidence angle of the level plane could be determined using the following equation:
i=α+β,
(13)
where
α=arccoscosϕncosϕdcosψnψd±sinϕnsinϕd,
(14)
β=arcsinRmoonsinα/Rem2sin2φemsin2φemRem/Rse+Rmoon22RemRmoonsinφemcosαsinφemRem/Rse1/2.
(15)

The Moon’s radius is denoted by Rmoon. The measured point’s selenographic latitude and longitude are represented by the values ψn and φn, respectively. ψd and φd are those of the subsolar point.

According to the analysis above, we can, therefore, calculate the ESI using the following equation:
I=S0sin2φemRem/Rsecos(α+β)Rem2sin2φem,
(16)
where α and β could be solved from Eqs. (14) and (15), respectively; ψd, φd, φem, Rem, and Rse could be obtained from the corresponding astronomical algorithms. Due to the Moon’s very tiny geocentric ecliptic latitude (0 ≤ φem ≤ 5.15°),19,20 the lunar-surface ESI could be simplified as the following equation:
Ilimφem0S0sin2φemRem/Rsecos(α+β)Rem2sin2φem=S0cos(α+β)Rse2.
(17)

The creation of a composite record using overlapping data for cross-calibration of readings from several radiometers is necessary for the reliable estimation of the solar constant, including HF/Nimbus, ACRIM1/SMM, ERBE/ERBS, ACRIM2/UARS, ACRIM3/ACRIM-Sat, VIRGO/SOHO, and SORCE/TIM.21–24 The model of PMOD developed by Fröhlich showed that the solar constant varies between 1361.8 and 1368.2 W·m−2. The average value of the solar constant is about 1366 W·m−2 by satellite observations. The range of the Sun–Earth distance is 0.98 AU to 1.02 AU.20 More than 100 W·m−2 is the variance in lunar-surface ESI brought on by variations in the Sun–Earth distance. The incidence angle of solar radiation spans from 0 to 90°. Consequently, the ESI of the lunar surface varies from 0 to 1425.7 W·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 Eq., it could be seen that the ESI is inclusive of the information of the longitude and latitude. Hence, the ESI is the embodiment of the location-dependent and seasonal variation characters. Figure 4 shows the ESI variation at different latitude regions from April 6, 1971, through April 5, 1972.

The Earth’s outgoing (infrared) radiation and the Earth’s reflection of solar radiation combine to create the earthshine (Fig. 5). The Moon, locked in synchronous rotation, only ever faces Earth from its nearside and always conceals its far side from it. 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
2πRem2E=(AS0+εσTe4)2πRe2,
(18)
where Re is the Earth’s radius and A is the terrestrial albedo. The following is a simplified version of Eq. (18):
E=(AS0+εσTe4)Re2Rem2.
(19)

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. A specific location on the nearside of the Moon is shielded from solar radiation and only gets terrestrial infrared radiation when the Earth lies between the Sun and the Moon, with the Moon just partially in the Earth’s shadow, that is, the reflected radiation is nearly zero. It is assumed that the average temperature of the Earth is 287 K. According to Goode et al.,25 the average albedo, based on data from the Big Bear Solar Observatory since December 1998, is 0.297 ± 0.005. With an average radius of the Earth of 6371 km, an average Earth–Moon distance of 384 402 km, and a solar constant of 1366 W·m−2, the infrared radiation received by the Moon is 0.099 W·m−2. Under most circumstances, an exact location on the nearside of the Moon receives between 0.099 and 0.201 W·m−2 of earthshine from the planet. In this paper, we choose 0.12 as the value of the earthshine.

Because it offers fundamental information for determining the thermal state of the lunar interior and is directly correlated with the amount of radiogenic elements, the lunar heat flow is significant. The heat flow at the Apollo 15 and 17 sites is measured to be 0.021 ± 0.003 W·m−2 and 0.014 ± 0.002 W·m−2, respectively.26 A global variation range was estimated to be from 0.02 W·m−2 to 0.04 W·m−2 based on the measurements made at the Apollo 15 and 17 sites and the microwave emission observations made from Earth.20 The two-layer model consists of a 2 cm surface layer with good thermal insulation and a subsurface layer with good thermal conductivity. The thermal conductivity is small within 2 cm of the surface layer, and the temperature changes dramatically, while the thermal conductivity is large but the temperature changes slowly below 2 cm. Except in the polar regions, the diurnal temperature variation extends to a depth of about 1 m, beyond which the temperature is essentially stable. At depth, the heat flux determines the temperature gradient.

According to the research by Wildey,27 the lunar surface layer’s reflectance ranges from 0.090 to 0.228, with an average of 0.125. It was estimated to be about 0.090–0.189 at the visible and infrared bands.28 Furthermore, Racca29 obtained the reflectivity of the lunar surface by fitting the Earth-based observation data, which is equal to 0.127. According to the data published by Wildey,27 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 2.3 K in the calculation of LST, when the reflectivity ranges from 0.102 to 0.127 or from 0.127 to 0.228. It is required to assume a constant value for the surface reflectivity in the specified range in order to simplify the calculations, and we use the reflectivity published by Racca29 to calculate the LST in this study.

In the thermal unsteady state, the emissivity is not equal to the absorptivity. Li et al.30 showed that a material has a stronger emission in thermal unsteady state than in thermal steady state. Therefore, the emissivity of lunar surface layer could not be deduced simply from the reflectivity. Hale and Hapke31 showed that the characteristics of thermal emission on the lunar surface are approximate to those of a blackbody. Its thermal emissivity is about 0.90–1.00.31–33 It is also essential to assume a constant value for the surface emissivity within the specified range in order to simplify the simulation. After averaging the values, ε = 0.94 has been taken into consideration for the simulation. With the mean values of I(t) = 1366 W·m−2 for lunar daytime and I(t) = 0 W·m−2 for lunar nighttime, E(t) = 0.12 W·m−2, and Q = 0.02 W·m−2, the LST could be computed by Eq. (1).

To verify the proposed model, in this section, the recorded temperatures at the Apollo 15 and 17 locations are compared with the simulated surface temperatures.

The constant parameters used in simulation are shown in Table I.

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.6 The surface temperatures at the Apollo 15 and 17 locations were recorded using thermocouples in cables positioned a few centimeters above the lunar surface, according to Langseth et al.7 Under the flux balance equation, these thermocouples are in radiative balance with the lunar surface, solar radiation, and space. When solving the flux balance equation, the site of Apollo 15 experiences a maximum temperature of 374 K and a minimum temperature of 92 K. At night, the temperature at the Apollo 17 site rises by roughly 10 K.34 

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. 6.

Figure 6 shows that 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 an atmosphere. At night, the variation of temperature is smooth, mainly controlled by the slow release of energy from the near-subsurface that was stored during the day by downward conduction.35 The simulation gives surface temperature with a maximum of 372.4 K and a minimum of 93.1 K, which are close to the measured maximum of 374 K and minimum of 92 K at the Apollo 15 site. The simulation also gives surface temperature with a maximum of 376 K and a minimum of 102.5 K at the Apollo 17 site, where the measured maximum surface temperature is 384 ± 6 K and the minimum surface temperature is 102 ± 2 K. Furthermore, the simulation represents that the temperature is increasing a little slower after dawn and decreasing a little slower after sunset than the measured temperature values at the Apollo 15 and 17 sites. Compared with Racca’s steady state temperature model, our model gives a commendably improved 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.29 

The calculated surface temperatures from our model are shown in Fig. 7, and the surface temperatures vary with time at different latitudes during an Earth year. An annual-seasonal variation can be observed, and the highest surface temperatures occur in late October, November, and December separately at high (>∼72°) latitudes, middle latitudes, and low (<∼20°) latitudes, respectively; the lowest surface temperatures occur in late July. It is also showed that the surface temperatures have an obvious seasonal variation at high latitudes. Moreover, the LSTs decrease with the increase in the latitude.

Owing to the lack of an 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 the “*” line show the maximum and minimum temperatures at the lunar nearside and farside separately (Fig. 8). 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 0.5 K lower than that at 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 in LST from the ESI, earthshine, and heat flow will be evaluated. A 0.12 W/m2 earthshine change from 0.12 to 0 W/m2 would result in an ∼0.5 K change in lunar surface nighttime temperature; a 0.02 W/m2 heat flow change from 0.02 to 0 W/m2 would result in an ∼0.09 K change in lunar surface nighttime temperature. Close to the equator, a 1322.5 W/m2 ESI change from 1322.5 to 0 W/m2 would result in a 179.4 K change in lunar surface daytime temperature.

In addition, the cooling rate is determined by the thermal diffusivity, k/ρc. The simulated temperature cooling curve shows a little slower cooling rate than the measured temperature cooling curve, which is possibly caused by the thermal diffusivity.

The lunar surface temperature is a particularly important quantity for explaining the thermal properties of regolith. Through the improved ESI model and thermal conduction model, we can calculate the surface temperature and lunar temperature profile. The thermophysical properties of the Moon can be calculated based on the theoretical model and the measured data. It will apply to provide a potential tool for age dating craters.36 In addition, the temperature of permanently shadowed regions in the polar regions has a correlation with water ice.37 

Lunar surface temperatures have a complicated relationship with the ESI, 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 that 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 in LST from the ESI, earthshine, and heat flow are evaluated. A 1322.5 W/m2 change in the ESI would lead to 179.4 K change in surface daytime temperature; a 0.12 and 0.02 W/m2 change in the earthshine and heat flow would lead to 0.5 and 0.09 K change 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 the Apollo 15 and 17 sites. The simulated surface temperature can reflect the change in LST during the lunar day and night. Once the accurate thermophysical parameters on a fixed location of the Moon are obtained, the LST would probably be 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 the lunar day, the surface temperature is determined by the solar radiation. It is obvious that the topography of the 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, (kρc)1/2, of the surface layers (k, ρ, 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 cannot 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.

This work was supported in part by the National Natural Science Foundation of China (NSFC) Program under Grant No. 42241138.

The authors have no conflicts to disclose.

Dan Zhang: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal). Wenchao Zheng: Formal analysis (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon request.

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