On the Solar Climate of the Moon and the Resulting Surface Temperature Distribution

Gerhard Kramm¹, Nicole Mölders², Martina Berger³, Ralph Dlugi³

¹Engineering Meteorology Consulting, Fairbanks, USA.
²Department of Atmospheric Sciences and Geophysical Institute, University of Alaska Fairbanks, Fairbanks, USA.
³Arbeitsgruppe Atmosphärische Prozesse (AGAP), Munich, Germany.

DOI: 10.4236/ns.2022.149034                

Abstract

The solar climate of our Moon is analyzed using the results of numerical simulations and the recently released data of the Diviner Lunar Radiometer Experiment (DLRE) to assess (a) the resulting distribution of the surface temperature, (b) the related global mean surface temperature <T_s>, and (c) the effective radiation temperature T_e often considered as a proxy for <T_s> of rocky planets and/or their natural satellites, where T_e is based on the global radiation budget of the well-known “thought model” of the Earth in the absence of its atmosphere. Because the Moon consists of similar rocky material like the Earth, it comes close to this thought model. However, the Moon’s astronomical features (e.g., obliquity, angular velocity of rotation, position relative to the disc of the solar system) differ from that of the Earth. Being tidally locked to the Earth, the Moon’s orbit around the Sun shows additional variation as compared to the Earth’s orbit. Since the astronomical parameters affect the solar climate, we predicted the Moon’s orbit coordinates both relative to the Sun and the Earth for a period of 20 lunations starting May 24, 2009, 00:00 UT1 with the planetary and lunar ephemeris DE430 of the Jet Propulsion Laboratory of the California Institute of Technology. The results revealed a mean heliocentric distance for the Moon and Earth of 1.00124279 AU and 1.00166376 AU, respectively. The mean geocentric distance of the Moon was 384792 km. The synodic and draconic months deviated from their respective means in a range of -5.7 h to 6.9 h and ±3.4 h, respectively. The deviations of the anomalistic months from their mean range between -2.83 d and 0.97 d with the largest negative deviations occurring around the points of inflection in the curve that represents the departure of the synodic month from its mean. Based on the two successive passages of the Sun through the ascending node of the lunar equator plane, the time interval between them corresponds to 347.29 days, i.e., it is slightly longer than the mean draconic year of 346.62 days. We computed the local solar insolation as input to the multilayer-force restore method of Kramm et al. (2017) that is based on the local energy budget equation. Due to the need to spin up the distribution of the regolith temperature to equilibrium, analysis of the model results covers only the last 12 lunations starting January 15, 2010, 07:11 UT1. The predicted slab temperatures, T_slab, considered as the realistic surface temperatures, follow the bolometric temperatures, T_bol, acceptably. According to all 24 DLRE datasets related to the subsolar longitude ø_ss, the global averages of the bolometric temperature amounts to T_bol=201.1k± 0.6K. Based on the globally averaged emitted infrared radiation of <F_IR>=290.5W·m^-2± 3.0W·m^-2 derived from the 24 DLRE datasets, the effective radiative temperature of the Moon is <T_{e, M}>=<T_bol>1/4=271.0k± 0.7K so that <T_bol>≅0.742T_{e, M}. The DLRE observations suggest that in the case of rocky planets and their natural satellites, the globally averaged surface temperature is notably lower than the effective radiation temperature. They differ by a factor that depends on the astronomical parameters especially on the angular velocity of rotation.

Keywords

Solar Climate, Temperature Inequality, Hölder’s Inequility, Global Radiation Budget, Local Radiation Budget, Global Energy Budget, Local Energy Budget, Global Albedo, Global Averaging, Effective Radiation Temperature, Surface Temperature, Slab Temperature, Multilayer-Force-Restore Method

1. INTRODUCTION

According to von Hann [1 , 2], the notion “solar climate” to which, for instance, Ptolemy’s climatic zones are related is based on the thought model of an Earth in the absence of its atmosphere. Von Hann stated [2]:

“If the surface of the Earth were occupied altogether by land, and if there were no surrounding atmosphere, the condition of our planet would be somewhat similar to that of the Moon at the present time. Under these conditions, the distribution of temperature over the Earth would depend solely upon the amount of heat received from the sun at any given place, and upon the loss of heat by radiation at that place. As these two factors would necessarily be the same at all points along the same parallel of latitude, the zones of equal temperature would coincide with the parallels of latitude. Even the presence of a vaporless atmosphere would interfere but little with this distribution of temperature, for only the absolute amounts of heat received at, and radiated from, the surface of the earth would thereby be affected. It is true that convectional currents would be produced under these conditions; but as there would be no reason for the more frequent occurrence of warm or cold air currents along some meridians than others, the distribution of temperature in zones bounded by the parallels of latitude would not thereby be interfered with.”

Thus, the solar climate (also called the mathematical climate [1 , 2]) of the Earth and its Moon essentially depends on astronomic conditions like the heliocentric distance, the obliquity of rotation axis with respect to the normal of the ecliptic plane, the angular velocity of the rotation, and the total solar irradiance (TSI), if scaled to 1 AU customarily called the solar constant. With respect to geological time scales, also changes in the precession of Earth’s rotation axis, and long-term variations of the eccentricity, obliquity, and precession of the Perihelion caused by the Sun, Moon and planets of our solar system must be addressed [3 - 9]. In the case of the Moon, the precession of the perigee as well as the precession of the lunar orbit’s longitude of the ascending node must be taken into account, which have periods of 8.85 years and 18.6 years, respectively.

The formula for the effective radiation temperature, $T_e$ , either of the Earth (with or without an atmosphere) or the Moon (or any other planet and natural satellites),

$T_e={\left(\frac{\left(1-{\alpha }_G\right)S}{4{\epsilon }_G\sigma }\right)}^{\frac{1}{4}}$ (1.1)

is crudely based on the solar climate. This formula is based on a global radiative equilibrium which means that the infrared radiation emanating from this conceptual Earth, ${\epsilon }_G\sigma T_e^4$ , is equal to the globally averaged absorbed solar radiation, $\left(1-{\alpha }_G\right)S/4$ [10]. Here, ${\alpha }_G$ is the global (or planetary) albedo in the solar range, S is the solar constant, ${\epsilon }_G$ is the global emissivity, and $\sigma =5.67\times {10}^{-8}\text{W}\cdot {\text{m}}^{-2}\cdot {\text{K}}^{-4}$ is Stefan’s constant. Note that the power law of Stefan [11] and Boltzmann [12] is only valid on a local scale. Applying it on a global scale notably disagrees with the prerequisites and assumptions on which the derivation of this power law is based. In the case of the Earth, the solar constant is about $S=1361\text{W}\cdot {\text{m}}^{-2}$ [13 - 16]. With the usual assumptions of ${\alpha }_G=0.30$ and ${\epsilon }_G=1.0$ , one obtains $T_e\approx 255\text{K}=-18˚\text{C}$ . Thus, the so-called natural atmospheric greenhouse effect is usually quantified by $\Delta T=\langle T_{ns}\rangle -T_e\approx 33\text{K}$ , where $\langle T_{ns}\rangle \approx 288\text{K}$ (e.g., [2 , 17 - 26]) is the globally averaged near-surface air temperature. Here, the angle brackets, $\langle \dots \rangle$ , define the global average (e.g., [19 , 27 - 29])

$\langle \psi \rangle =\frac{1}{4\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\pi }{\int }}\psi \left(\theta ,\phi \right)\sin \theta \text{d}\theta \text{d}\phi }}=\frac{1}{2}{\displaystyle \underset{0}{\overset{\pi }{\int }}\bar{\psi }\left(\theta \right)\sin \theta \text{d}\theta }$ (1.2)

with the zonal average (e.g., [28 - 32]),

$\bar{\psi }\left(\theta \right)=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{2\pi }{\int }}\psi \left(\theta ,\phi \right)\text{d}\phi }$ (1.3)

where $\psi \left(\theta ,\phi \right)$ is a field quantity like the solar insolation, $F_S\left(\theta ,\phi \right)$ , the absorbed solar irradiance, $Q\left(\theta ,\phi \right)$ , the infrared irradiance, $F_{IR}\left(\theta ,\phi \right)$ , and the surface temperature, $T_s\left(\theta ,\phi \right)$ .

The assumption of a global albedo of about ${\alpha }_G=0.30$ , however, is far from reality because this value is related to the entire Earth-atmosphere system. For ${\alpha }_G=0.30$ , the infrared radiation emitted to space would be that at the top of the atmosphere (TOA) of about $\langle F_{IR}\rangle \cong 238.2\text{W}\cdot {\text{m}}^{-2}$ . The cloud cover primarily contributes to this value of the global albedo, but a cloud cover cannot exist in the thought model of the Earth in the absence of its atmosphere. Furthermore, the global emissivity of such a conceptual Earth is unknown. Both quantities might be related to those of the Earth’s Moon, as done by Kramm et al. [29].

Since, however, the thought model of an Earth in the absence of an atmosphere eludes observation, there are some assumptions that can be combined at will. Inserting, for instance, ${\epsilon }_G=0.8$ as assumed by Schack [33], and ${\alpha }_G=0.07$ as suggested by Budyko [34] for the Earth in the absence of its atmosphere into Equation (1.1) would provide $T_e=289\text{K}$ . Thus, we would obtain $\Delta T=\langle T_{ns}\rangle -T_e\approx -1\text{K}$ .

The effective radiation temperature of the Earth either with or without an atmosphere is only a synonym for the global average of the infrared radiation emitted to space. It is a measure of the intensity of the radiation emanating from the Earth and, therefore, says nothing about the existing temperature distribution [35]. It was considered, for instance, by Defant and Obst [36], Lenard [37], Möller [38], and later adopted by many others like Hansen et al. [24] to quantify the so-called greenhouse effect. However, in the case of the Earth in the absence of its atmosphere, $T_e$ would only correspond to a globally averaged surface temperature if the surface temperature were uniformly distributed, which is, by far, not the case. A uniform distribution of the surface temperature would only exist in the trivial case that the solar constant for the planet or natural satellite would be zero. As illustrated in Figures 1-3, the distribution of the surface temperature on a planet or a natural satellite (like Earth’s Moon or Jupiter’s Galilean moon Io) in the absence of an atmosphere is non-uniform. As outlined by von Hann, this distribution of the surface temperature is mainly governed by the solar insolation defined as the flux of solar radiation per unit of horizontal area for a given location [8].

Figure 1. Distributions of (a) the bolometric temperature obtained from the Lunar Reconnaissance Orbiter Diviner Lunar Radiometer Experiment for the subsolar longitude ${\varphi }_{ss}=0˚E$ according to Williams et al., [39], and the daily mean values of (b) the slab temperature, (c) the solar insolation at Moon’s surface, and (d) the absorbed solar radiation calculated for twelve synodic months starting with TDB = 2455211.8 (January 15, 2010, 11:07 UT1, New Moon) and the latitudes ranging from $\varphi =90˚$ (North Pole) to $\varphi =-90˚$ (South Pole) at an equidistant distance of $\Delta \varphi =5˚$ , where 144 values per day were used for daily averaging. The numerical simulations were performed using the multilayer-force-restore method that is based on Equation (2.1) and a multilayer numerical model that is based on Equation (2.6) for calculating the heat transfer in the regolith (adopted from Kramm et al. [29]).

Note that the notion “terrestrial day” used in Figure 1 means the length of the day of 86,400 s = 24 h. It is about 236 s longer than the Earth’s sidereal rotation period. Hereafter, we only use “day” (d), where 365.25 d = 1 Julian year.

Figure 2. Daily mean values of (a) the slab temperature according to Equation (2.1), and (b) the solar insolation at the surface of the Earth in the absence of the atmosphere which were calculated for all days of a year, starting with TDB = 2455197.5 (January 1, 2010, 00:00 UT1) and the latitudes ranging from $\varphi =90˚$ (North Pole) to $\varphi =-90˚$ (South Pole) at an equidistant distance of $\Delta \varphi =5˚$ , where 144 values per day were used for daily averaging. The numerical simulations were performed using the so-called multilayer-force-restore method as well (adopted from Kramm et al. [29]).

Figure 3. Map of daytime brightness temperature in the PPR 27 μm filter, superimposed on an SSI map of Jupiter’s Galilean moon Io, from the 31IPDGTM_01 observation. Contour interval is 5 K. Approximate location of the sub-solar point is marked with an * (adopted from Rathbun et al. [51]).

The concept of effective radiation temperature was originally developed for stars like our Sun (e.g., [35 , 40 - 42]). It seems that this concept is unsuitable for planets and their natural satellites. The solar irradiance, F, reaching the TOA or—if an atmosphere plays no role—the surface of either a planet or a natural satellite in our solar system at the sub-solar point is given by

$F={\left(\frac{r_{Sun}}{r}\right)}^2{F}_{Sun}$ (1.4)

where $F_{Sun}$ is the solar emittance [8 , 9 , 43 , 44], $r_{Sun}\cong 6.963\times {10}^5\text{km}$ [45] is the visible radius of the Sun, and r is the actual heliocentric distance of either the planet or the natural satellite. Formula (1.4) is based on the fact that the radiant power emitted by the Sun is kept constant when the solar radiation is propagating through the space because of energy conservation principles in the absence of an intervening medium [8 , 46 , 47]. This radiant power (also called the luminosity) is given

by

$L=4\pi r_{Sun}^2{F}_{Sun}=4\pi r^{2}F=3.828\times {10}^{26}\text{W}$ (1.5)

Inserting the mean heliocentric distance, $r_0$ , of a planet or a natural satellite into Equation (1.4) provides the respective solar constant

$S={\left(\frac{r_{Sun}}{r_0}\right)}^2{F}_{Sun}$ (1.6)

Combining formulae (1.4) and (1.6) yields

$F={\left(\frac{r_0}{r}\right)}^{2}S$ (1.7)

The solar constant of the Earth of $S\cong 1361\text{W}\cdot {\text{m}}^{-2}$ mentioned before is related to the mean heliocentric distance of $r_0\cong 1.496\times {10}^8\text{km}$ (nearly 1 AU) [14 , 48 - 50], where the actual heliocentric distance of the Earth (strictly spoken the Earth-Moon barycenter, EMB) ranges from $r\cong 1.471\times {10}^8\text{km}$ at the Perihelion to $r\cong 1.521\times {10}^8\text{km}$ at the Aphelion.

Inserting Equation (1.6) into Equation (1.1) provides

$T_e={\left(\frac{\left(1-{\alpha }_G\right)F_{Sun}}{{\epsilon }_G\sigma }\right)}^{\frac{1}{4}}{\left(\frac{r_{Sun}}{2r_0}\right)}^{\frac{1}{2}}={\left(\frac{F_{Sun}}{\sigma }\right)}^{\frac{1}{4}}{\left(\frac{1-{\alpha }_G}{{\epsilon }_G}\right)}^{\frac{1}{4}}{\left(\frac{r_{Sun}}{2r_0}\right)}^{\frac{1}{2}}$ (1.8)

Introducing the effective radiation temperature of the Sun, $T_{Sun}={\left(F_{Sun}/\sigma \right)}^{1/4}$ [41], assuming ${\epsilon }_G=1$ and replacing $r_0$ by the orbital semi-major axis, a yield [52]

$T_e\cong T_{Sun}{\left(1-{\alpha }_G\right)}^{\frac{1}{4}}{\left(\frac{r_{Sun}}{2a}\right)}^{\frac{1}{2}}$ (1.9)

The formula should also serve for determining the effective radiation temperature of Earth analogs. Furthermore, the luminosity (Equation (1.5)) is used to define the scaled semi-majsor axis by [52]

$a_s=\frac{a}{\sqrt{L}}={\left(4\pi S\right)}^{-\frac{1}{2}}$ (1.10)

Unfortunately, the effective radiation temperature is not unambiguous because non-uniform distributions of the surface temperature are also compatible with the global radiation balance. Gerlich and Tscheuschner [53], for instance, derived the following formula for the globally averaged surface temperature of a radiation-exposed static globe in the absence of its atmosphere:

$\langle T_s\rangle =\frac{2^{3/2}}{5}{T}_e\cong 0.566T_e$ (1.11)

As assumed by these authors, $T_s\left(\theta ,\phi \right)$ is the surface temperature that is based on the local radiation balance given by

$\left(1-\alpha \left({\Theta }_0,\theta ,\phi \right)\right)F\cos {\Theta }_0=\epsilon \left(\theta ,\phi \right)\sigma T_s^4\left(\theta ,\phi \right)$ (1.12)

where $Q=\left(1-\alpha \left({\Theta }_0,\theta ,\phi \right)\right)F\cos {\Theta }_0$ is the absorbed solar radiation, $\alpha \left({\Theta }_0,\theta ,\phi \right)$ is the integral albedo of the solar range, ${\Theta }_0$ is the local zenith distance of the Sun’s center, and the infrared radiation emitted by the regolith in the close vicinity of the surface is given by the power law of Stefan and Boltzmann $F_{IR}=\epsilon \left(\theta ,\phi \right)\sigma T_s^4\left(\theta ,\phi \right)$ , where $\epsilon \left(\theta ,\phi \right)$ is the integral relative emissivity and $\sigma$ is Stefan’s constant. As both the Earth and the Moon are considered as spheres, the location is simply characterized by the zenith and azimuthal angles $\theta$ and $\phi$ , respectively. The zenith angle, $\theta$ , ranges from 0 (North Pole) to π (South Pole), and the azimuthal angle, $\phi$ , ranges from 0 to 2π . The global average of the absorbed solar radiation is given by

$\langle Q\rangle =\langle F\cos {\Theta }_0\rangle -\langle \alpha \left({\Theta }_0,\theta ,\phi \right)F\cos {\Theta }_0\rangle =\left(1-{\alpha }_G\right)\langle F\cos {\Theta }_0\rangle$ (1.13)

where

${\alpha }_G=\frac{\langle \alpha \left({\Theta }_0,\theta ,\phi \right)F\cos {\Theta }_0\rangle }{\langle F\cos {\Theta }_0\rangle }$ (1.14)

defines the globally averaged albedo in the solar range. Generally, ${\alpha }_G\ne \langle \alpha \left({\Theta }_0,\theta ,\phi \right)\rangle$ , except for $\alpha \left({\Theta }_0,\theta ,\phi \right)=const$ . Note that $\langle F\cos {\Theta }_0\rangle \cong S/4$ . The global average of the emitted infrared radiation is given by

$\langle F_{IR}\rangle =\langle \epsilon \left(\theta ,\phi \right)\sigma T_s^4\left(\theta ,\phi \right)\rangle ={\epsilon }_G\sigma \langle T_s^4\left(\theta ,\phi \right)\rangle$ (1.15)

where

${\epsilon }_G=\frac{\langle \epsilon \left(\theta ,\phi \right)T_s^4\left(\theta ,\phi \right)\rangle }{\langle T_s^4\left(\theta ,\phi \right)\rangle }$ (1.16)

defines the globally averaged emissivity, where ${\epsilon }_G\ne \langle \epsilon \left(\theta ,\phi \right)\rangle$ , except for $\epsilon \left(\theta ,\phi \right)=const$ .

Choosing ${\alpha }_G=0.30$ , and ${\epsilon }_G=1.0$ , as done for instance by Gerlich and Tscheuschner [53], provides $\langle T_s\rangle \cong 144.3\text{K}$ . Even though this global average of the surface temperature drastically differs from effective radiation temperature, the globally averaged emitted infrared radiation is $\langle Q\rangle =\langle F_{IR}\rangle =\sigma \langle T_s^4\left(\theta ,\phi \right)\rangle \cong 238\text{W}\cdot {\text{m}}^{-2}$ . They denoted $\langle T_s\rangle$ as the “physical temperature.” This notion, however, is infelicitous. As a global average, $\langle T_s\rangle$ has no physical meaning. Its sole purpose is to compare it with $\langle T_{ns}\rangle$ because both quantities are globally averaged in the same manner. With respect to their results, Gerlich and Tscheuschner argued that the average temperatures are considerably lower than the absolute temperature’s fourth root of the averaged fourth power expressed by their temperature inequality [53]

$\langle T_s\rangle \le \sqrt[4]{\langle T_s^4\rangle }$ , (1.17)

verified by them with the aid of Hölder’s [54] inequality, but formulated for integrals [55].

Kramm et al. [29] obtained $\langle T_s\rangle \cong 148.4\text{K}$ , $\langle Q\rangle \cong 279.7\text{W}\cdot {\text{m}}^{-2}$ , and $\langle F_{IR}\rangle \cong 279.6\text{W}\cdot {\text{m}}^{-2}$ for an obliquely rotating Earth in the absence of its atmosphere when using Equation (1.12), $\epsilon \left(\theta ,\phi \right)=0.98$ , and Keihm’s [56] empirical formula rearranged to

$\alpha \left({\Theta }_0\right)={\alpha }_0+{\left(\frac{{\Theta }_0}{45˚}\right)}^3\left(a+b{\left(\frac{{\Theta }_0}{45˚}\right)}^5\right)$ (1.18)

where ${\alpha }_0=0.10$ is the normal albedo, $a=0.045$ and $b=5.47\times {10}^{-4}$ are empirical parameters (with exception of b, all other values are from observations of the Diviner Lunar Radiometer Experiment (DLRE) [57 , 58] aboard NASA’s Lunar Reconnaissance Orbiter (LRO) [59 - 61]). Because ${\alpha }_G=0.178$ , the effective radiation temperature is $T_e\cong 266.4\text{K}$ . This means that the result of Gerlich and Tscheuschner [53] for a non-rotating Earth in the absence of its atmosphere holds even for an obliquely rotating globe if atmospheric effects were negligible.

Equation (1.12) expresses the solar climate in its historical sense, i.e., the distribution of temperature over the Earth in the absence of its atmosphere would depend solely upon the amount of heat received from the Sun at any given place, and upon the loss of heat by radiation at that place [1 , 2]. However, as already demonstrated by Wesselink [62] in the case of the Moon, the local radiation balance fails completely during the nighttime because it would provide $T_s=0\text{K}$ . It is therefore essential to expand the concept of solar climate in such a way that such a model artifact is generally excluded. Consequently, the soil heat flux has to be considered as well, as done, for instance, by Wesselink [62], Jaeger [63], Cremers et al. [64], Mitchell and de Pater [65], Vasavada et al. [58 , 66], Bauch et al. [67 , 68], Hu et al. [69], and Kramm et al. [29].

An Earth in the absence of an atmosphere, however, is beyond the scope of observation. Consequently, at least, the plausibility of this thought model must be assessed in another way. In our solar system, for instance, the planet Mercury and our Moon are suitable approximations for a rocky planet without an atmosphere because the densities of their atmospheres are very low. Therefore, these Earth analogs come very close to the thought model of an Earth in the absence of its atmosphere. However, their astrometric features considerably differ from those of the Earth.

The Earth’s period of rotation is 23.9345 hours, which corresponds to an angular velocity of rotation of ${\omega }_E\cong 7.292\times {10}^{-5}{\text{s}}^{-1}$ . Since the sidereal period of rotation of the Moon is 655.73 hours, which corresponds to an angular velocity of ${\omega }_{Mo}\cong 2.662\times {10}^{-6}{\text{s}}^{-1}$ , the Moon rotates 27.4 times slower than the Earth. Mercury’s sidereal period of rotation is 1407.6 hours and consequently the angular velocity of its rotation is ${\omega }_{Me}\cong 1.240\times {10}^{-6}{\text{s}}^{-1}$ . This means that Mercury rotates 58.8 times slower than the Earth and 2.15 times slower than the Moon.

The mean heliocentric distance of Mercury is only $r_{0,Me}\approx a_{Me}=5.791\times {10}^7\text{km}=0.3871\text{AU}$ leading to a solar constant of

$S_{Me}={\left(\frac{r_{0,E}}{r_{0,Me}}\right)}^{2}S\cong 9083\text{W}\cdot {\text{m}}^{-2}$ (1.19)

In addition, Mercury’s orbit shows an extremely high eccentricity of $e_{Me}\cong 0.2056$ , while that of the Earth orbit is only $e_E\cong 0.0167$ . Since Mercury’s heliocentric distance ranges between 4.600 × 10⁷ km (Perihelion) and 6.982 × 10⁷ km (Aphelion), its total solar irradiance (TSI) varies from $F_{Me}\approx 14395\text{W}\cdot {\text{m}}^{-2}$ at Perihelion to $F_{Me}\approx 6249\text{W}\cdot {\text{m}}^{-2}$ at Aphelion. A special feature of Mercury is that the angular velocity of the orbit near the Perihelion slightly exceeds that of the rotation, which leads to a secondary sunrise (e.g., [66 , 68 , 70 , 71]). Against this background, the Moon is preferable for testing the plausibility of the thought model of an Earth in absence of its atmosphere. Further reasons are the existence of (a) the in-situ temperature measurements carried out with the help of thermocouples as part of the heat flow experiments of the Apollo Lunar Surface Experiment Package (ALSEP) at the Apollo 15 landing site Hadley Rille/Apennine Mountains [72] and the Apollo 17 landing site Taurus-Littrow [73] and (b) the bolometric temperatures provided by the DLRE [57], available to us since April 2018. Nonetheless, the Mercury Radiometer and Thermal Infrared Spectrometer (MERTIS) that is part of the BepiColombo Mission launched in October 2018 [68 , 74] may provide a reliable database for evaluating model results regarding Mercury’s solar climate.

Kramm et al. [29] used their multilayer-force-restore method to compute the global average of the Moon’s surface temperature and in a further step that of the Earth in the absence of its atmosphere. Kramm et al. obtained for the Moon $\langle T_{slab}\rangle \cong 197.9\text{K}$ and for the Earth in the absence of its atmosphere $\langle T_{slab}\rangle \cong 220.7\text{K}$ . Thus, the outcome notably differs. This difference can be explained by the 27.4 times higher angular velocity of the Earth compared to that of the Moon, i.e., the response time of the emitted infrared radiation with respect to the absorbed solar radiation causes different effects [29]. Kramm et al.

obtained for the Moon ${\langle T_{slab}^4\left(\theta ,\phi \right)\rangle }^{1/4}\cong 266.4\text{K}$ and for the Earth ${\langle T_{slab}^4\left(\theta ,\phi \right)\rangle }^{1/4}\cong 266.5\text{K}$ , i.e., ${\langle T_{slab}^4\left(\theta ,\phi \right)\rangle }^{1/4}$ and $T_e$ substantially agree with each other, but in contrast to Equation (1.11), $\langle T_{slab}\rangle \cong 0.743{\langle T_{slab}^4\left(\theta ,\phi \right)\rangle }^{1/4}$ is valid for the obliquely rotating Moon and $\langle T_{slab}\rangle \cong 0.828{\langle T_{slab}^4\left(\theta ,\phi \right)\rangle }^{1/4}$

for the obliquely rotating Earth in the absence of its atmosphere. These results confirm the temperature inequality of Gerlich and Tscheuschner [53].

Kramm et al. [29] preliminarily validated their model results obtained for the Moon by comparing the zonal mean bolometric temperatures of the DLRE published by Williams et al. [39] for numerous parallels of latitude. As mentioned before, these DLRE datasets are meanwhile available. The goals of our paper are, therefore, (a) to expand the notion “solar climate” to predict the related distribution of the surface temperature in a realistic manner, (b) to evaluate the model results of Kramm et al. on the basis of the 24 DRLE datasets, (c) to compare the globally averaged surface temperature with the effective radiation temperature, and (d) to assess the temperature inequality of Gerlich and Tscheuschner [53] in the case of Earth analogs, where we exemplarily consider our Moon as a test bed.

2. THE NUMERICAL MODEL FOR THE MOON REGOLITH

Since the soil heat flux has to be considered in deriving the surface temperature [53], Kramm et al. [29] derived the following local energy budget equation for a thin slab of the regolith adjacent to the surface of either the Moon or the Earth in the absence of its atmosphere:

$\vartheta \frac{\text{d}}{\text{d}t}{\langle c\rho T_{slab}\rangle }_V=\left(1-\alpha \left({\Theta }_0,\theta ,\phi \right)\right)F\cos {\Theta }_0-\epsilon \left(\theta ,\phi \right)\sigma T_{slab}^4\left(\theta ,\phi \right)-H_{sl}\left(\theta ,\phi \right)$ (2.1)

Here, t is time, $T_{slab}$ , $\rho$ , and c, are the temperature, bulk density, and specific heat of this slab, respectively. The soil volume average is defined by

${\langle c\rho T_{slab}\rangle }_V=\frac{1}{V_{slab}}{\displaystyle \underset{V_{slab}}{\int }c\rho T_{slab}\text{d}V}$ (2.2)

The volume $V_{slab}$ is given by $V_{slab}=C_{slab}\vartheta$ , where $C_{slab}$ is the cross section, and $\vartheta =2\text{cm}$ is the thickness of this slab. Thus, the temperature $T_{slab}$ represents the temperature of the slab adjacent to the surface. Assuming that $T_{slab}$ , $\rho$ , and c are homogeneously distributed in this thin layer leads to

$R_t\frac{\text{d}{T}_{slab}}{\text{d}t}=\left(1-\alpha \left({\Theta }_0,\theta ,\phi \right)\right)F\cos {\Theta }_0-\epsilon \left(\theta ,\phi \right)\sigma T_{slab}^4\left(\theta ,\phi \right)-H_{sl}\left(\theta ,\phi \right)$ (2.3)

where $R_t=c\rho \vartheta$ is the thermal inertial coefficient. Furthermore, $H_{sl}\left(\theta ,\phi \right)$ is the vertical component of the soil heat flux density expressed by the one-dimensional form of Fourier’s law of heat conduction (e.g., [62 - 66 , 75 - 77]),

$H_{sl}\left(\theta ,\phi \right)=-k_h\left(\theta ,\phi \right){\frac{\partial T_{sl}}{\partial z}|}_{\theta ,\phi }$ (2.4)

Here, $T_{sl}$ is the soil temperature, and $k_h\left(\theta ,\phi \right)$ is the thermal conductivity. The soil heat flux density characterizes the exchange of the slab with soil layers below. Henceforth, a flux density is called a flux for ease of readability. The difference between the absorbed solar radiation and the emitted infrared radiation mainly governs the direction of $H_{sl}\left(\theta ,\phi \right)$ .

Under steady-state conditions, the left-hand side of Equation (2.3) is zero leading to

$\left(1-\alpha \left({\Theta }_0,\theta ,\phi \right)\right)F\cos {\Theta }_0-\epsilon \left(\theta ,\phi \right)\sigma T_s^4\left(\theta ,\phi \right)-H_{sl}\left(\theta ,\phi \right)=0$ (2.5)

This means that the slab, characterized by $T_{slab}$ , $\rho$ , c, and $\vartheta$ , no longer occurs, and $T_{slab}$ is replaced by the “surface temperature” $T_s$ . Wesselink [62], Jaeger [63], Cremers et al. [64], Mitchell and de Pater [65], Vasavada et al. [58 , 66], Bauch et al. [67 , 68], for instance, used Equation (2.5) together with Equation (2.4) to compute the surface temperature for various areas on the Moon.

Kramm et al. [29] used Equation (2.3) together with Equation (2.4) to predict the distributions of slab temperatures for both the Earth’s Moon and the Earth in the absence of its atmosphere. A numerical multilayer model for the regolith based on

$\rho c\frac{\partial T_{sl}}{\partial t}=\frac{\partial }{\partial z}\left(k_h\frac{\partial T_{sl}}{\partial z}\right)$ (2.6)

was used to predict the both the temperature and the heat flux in the layer of the regolith below the slab. Kramm et al. used 16 levels in their computations and a maximum depth of z_r = 3.20 m, at which T_sl.r, is considered as time-invariant, but dependent on latitude [66].

For the bulk density and the thermal conductivity, Kramm et al. considered the formulae of Vasavada et al. [58],

$\rho \left(z\right)={\rho }_b-\left({\rho }_b-{\rho }_t\right)\exp \left(-\frac{z}{0.06}\right)$ (2.7)

where ${\rho }_t=1300\text{kg}\cdot {\text{m}}^{-3}$ and ${\rho }_b=1800\text{kg}\cdot {\text{m}}^{-3}$ are the bulk densities close to the surface and at the depth $z_r$ , respectively, and

$k_h\left(z,T\right)=k_{h,b}-\left(k_{h,b}-k_{h,t}\right)\exp \left(-\frac{z}{0.06}\right)+k_{h,t}\chi {\left(\frac{T}{T_{350}}\right)}^3$ (2.8)

with $k_{h,t}=\text{6}\text{.0}\times {\text{10}}^{-4}\text{W}\cdot {\text{m}}^{-1}\cdot {\text{K}}^{-1}$ , $k_{h.b}=7.0\times {10}^{-3}\text{W}\cdot {\text{m}}^{-1}\cdot {\text{K}}^{-1}$ , and $\chi =2.7$ . The heat capacity was calculated by

$c=\{\begin{array}{l}-\text{23}\text{.17}+\text{744}\text{.5}\left(\frac{T}{T_{350}}\right)+\text{1839}{\left(\frac{T}{T_{350}}\right)}^2-\text{3160}{\left(\frac{T}{T_{350}}\right)}^3+\text{1449}{\left(\frac{T}{T_{350}}\right)}^4\text{for}T\le T_{350}\\ \text{1009}-\text{5307}\exp \left(-3.5\left(\frac{T}{T_{350}}\right)\right)\text{for}T>T_{350}\end{array}$ (2.9)

The formula for $T\le T_{350}$ is based on the analysis of lunar soils samples from the lunar landing sites Fra Mauro (Apollo 14), Hadley-Apeninne Base (Apollo 15), and Descartes Highlands (Apollo 16) by Hemingway et al. [78], but their results are normalized by $T_{350}$ . Wechsler et al. [79] recommended an exponential function for $T>T_{350}$ . For a detailed discussion of the numerical procedures and results provided by this multilayer force-restore method see Kramm et al. [29]. As mentioned before, Figures 1(b)-(d) and Figure 2 are based on the results provided by this multilayer force-restore method.

3. ASTROMETRIC ASPECTS

The local energy budget equation for a thin slab of the regolith adjacent to the surface of either the Moon or the Earth in the absence of its atmosphere given by Equations (2.1), (2.3), and (2.5) as well as the local radiation budget expressed by Equation (1.12) requires the solar input. For computing the TSI, we need Equation (1.7), the solar constant, and the local zenith distance of the Sun’s center.

In the case of the Earth, $\cos {\Theta }_0$ can be determined using the rules of spherical trigonometry (e.g., [8 , 9 , 36 , 80 - 85])

$\begin{array}{c}\cos {\Theta }_0=\sin \varphi \sin {\delta }_S+\cos \varphi \cos {\delta }_S\cos h\\ =\cos \theta \sin {\delta }_S+\sin \theta \cos {\delta }_S\cos h\end{array}$ (3.1)

Here, ${\delta }_S$ is the declination of the Sun, $\varphi$ is latitude, and h is the hour angle with respect to the local meridian. We use a spherical coordinate frame with its origin in the center of a planet or a natural satellite, the zenith angle, $\theta$ , is counted with respect to the body’s rotation axis, which shows in the direction of the Northern Celestial Hemisphere. The declination of the Sun can be determined using

$\sin {\delta }_S=\sin \epsilon \sin \lambda =\sin \epsilon \sin \left(\upsilon +\varpi \right)$ (3.2)

where $\epsilon$ is the obliquity of the ecliptic, and $\lambda =\upsilon +\varpi$ is the true longitude of the Earth counted counterclockwise from the vernal equinox (e.g., [8 , 9 , 84 , 86 , 87]), $\upsilon$ is the true anomaly, i.e., the positional angle of the Earth on its orbit counted counterclockwise from the Perihelion, and $\varpi$ is the longitude of the Perihelion counted counterclockwise from the moving vernal equinox of the Northern Hemisphere. The declination ${\delta }_S$ ranges from ${\delta }_S={23}^{\circ }26\prime 21″\text{S}$ (Tropic of Capricorn; $\lambda =3\pi /2$ ) to ${\delta }_S={23}^{\circ }26\prime 21″\text{N}$ (Tropic of Cancer; $\lambda =\pi /2$ ), and h ranges from −H to H, where H represents the half-day, i.e., from sunrise to solar noon or solar noon to sunset. The half-day must fulfill the condition $H<\pi$ , where $\pi$ corresponds to 12 hours. This condition is not fulfilled for those points within the polar domes delimited by the polar circles for which the Sun does not set within 24 hours, because then $\cos {\Theta }_0=0$ does not hold [81 , 82]. Therefore, the condition $H=\pi$ is that of the polar circle, i.e., $\varphi =-{66}^{\circ }33\prime 39″$ (Antarctic circle) and $\varphi ={66}^{\circ }33\prime 39″$ (Arctic circle). Furthermore, the obliquity is given by $\epsilon =\bar{\epsilon }+\Delta \epsilon$ , where the mean obliquity of date,

$\bar{\epsilon }=84381^{″}.448-46^{″}.815T-0^{″}.00059T^2+0^{″}.001813T^3$ (3.3)

is adopted from Folkner et al. [88] and Park et al. [89]. The nutation in the obliquity is given, for instance, by $\Delta \epsilon =9{.}^{″}205348\cos \Omega$ [88], where the ascending node of Moon’s orbit on the ecliptic reads [88 , 89]

$\Omega ={125}^{\circ }02^{\prime }40^{″}.280-{1934}^{\circ }08^{\prime }10^{″}.549T+7^{″}.455T^{\text{2}}+0^{″}.008\text{}{T}^{\text{3}}$ (3.4)

Here, T is the TDB time in centuries with respect to J2000.0, where the Julian century corresponds to 36525 d.

For the Moon, ${\delta }_S$ has to be replaced by the selenographic latitude, $b_S$ , of the Sun. In accord with Taylor et al. [90], Kramm et al. [29] computed the selenographic longitude, $l_S$ , and the selenographic latitude, $b_S$ , of the Sun using

$\tan \left(l_S+L_M-\Omega \right)=\frac{\cos I\cos {\beta }_H\sin \left({\lambda }_H-\Omega -\Delta \psi \right)-\sin I\sin {\beta }_H}{\cos {\beta }_H\cos \left({\lambda }_H-\Omega -\Delta \psi \right)}$ (3.5)

and

$\sin b_S=-\sin I\cos {\beta }_H\sin \left({\lambda }_H-\Omega -\Delta \psi \right)-\cos I\sin {\beta }_H$ (3.6)

where the selenographic colongitude is ${90}^{\circ }-l_S$ . Here, $I=5553^{″}.6\cong \text{1}{\text{.54267}}^{\circ }$ is the inclination of the ecliptic to the mean lunar equator adopted from Newhall and Williams [91], $L_M$ is the mean longitude of the Moon adopted from Simon et al. [92], and $\Delta \psi$ is the nutation in longitude given, for instance, by $\Delta \psi =-17^{″}.206262\sin \Omega$ [88]. Kramm et al. [29], however, used the data provided by the planetary and lunar ephemeris DE430 of the Jet propulsion Laboratory (JPL), California Institute of Technology. Since 2020, JPL’s planetary and lunar ephemeris DE440 is available [89]. Therefore, we confirmed the astrometric results of Kramm et al. from 2017 using DE440. In some cases, we compared our astrometric results with those from the JPL Horizons on-line solar system (https://ssd.jpl.nasa.gov/horizons/) that are based on DE441 because these results are provided by independent calculations.

The heliocentric ecliptic latitude and longitude of the Moon, ${\beta }_H$ and ${\lambda }_H$ , are given by

$\sin {\beta }_H=\frac{Z_{SM}}{d_{SM}}$ (3.7)

and

$\tan {\lambda }_H=\frac{Y_{SM}}{X_{SM}}$ (3.8)

Here, $d_{SM}=\left|d_{SM}\right|$ is the length of the heliocentric vector to the Moon, $d_{SM}=\left(X_{SM},Y_{SM},Z_{SM}\right)$ with the coordinates $X_{SM}$ , $Y_{SM}$ , and $Z_{SM}$ . Kramm et al. [29] also used JPL’s DE430 to compute the heliocentric distances of the Moon, $r_M$ , and the Earth, $r_E$ , the declination, ${\delta }_S$ , and the selenographic latitude, $b_S$ , of the Sun, respectively.

Results for these quantities obtained for about twenty synodic months (about 591 days) starting May 24, 2009, 00:00 UT1 (TDB = 2454975.5), denoted hereafter as Period I, are illustrated in Figure 4(a) and Figure 5, respectively. Based on Equation (1.7) and the values of $r_M$ , $r_E$ , and S, the corresponding TSI reaching either the Moon or the Earth are predicted as well (Figure 4(b)). The results illustrated in Figure 4(a) lead to a mean heliocentric distance for the Moon of 1.0012479 AU and for the Earth of 1.00166376 AU, where 1 AU = 149597870.700 km. Since the multilayer-force-restore method has to be spun up to equilibrium prior to analysis of the results [93], only the results provided by it for the last twelve synodic months starting January 15, 2010, 07:11 UT1 (TDB = 2455211.8, New Moon), hereafter denoted as Period II, were analyzed by Kramm et al. [29].

Figure 4. Variation of (a) the heliocentric distances of the Moon, $r_M$ , and the Earth, $r_E$ , during the Period I, derived from the data provided by JPL’s ephemeris DE430, and (b) the TSI reaching the sub-solar point either of the Earth or Moon as obtained with Equation (1.7) for $S=1361W\cdot m^{-2}$ .

Figure 5. Variation of (a) the selenographic latitude, $b_S$ , and (b) the selenographic longitude, $l_S$ of the Sun during the Period I. The results are based on the data provided by JPL’s ephemeris DE430.

The synodic month (also referred to as a lunation) is defined as the interval between two consecutive New Moons. It is nearly 2.21 days longer than the lunar orbital period with respect to the fixed stars called the sidereal month (see Table 1). While the Moon revolves around Earth, both objects also progress in orbit around the Sun. After completing one revolution with respect to the fixed stars, the Moon must continue a little farther along its orbit to catch up to the same position it started from relative to the Sun and Earth [94].

At the time of the New Moon, solar radiation reaches a local maximum. Thus, the calculation of the TSI requires an accurate determination of the New Moon. It occurs when the geocentric ecliptic longitudes of the Sun ( ${\lambda }_S$ ) and the Moon ( ${\lambda }_M$ ) are the same (e.g., [94 - 96]), customarily expressed by $f\left(t\right)=0$ , where $f\left(t\right)={\lambda }_M-{\lambda }_S$ for New Moon $f\left(t\right)={\lambda }_M-{\lambda }_S-{90}^{\circ }$ for the first quarter, $f\left(t\right)={\lambda }_M-{\lambda }_S-{180}^{\circ }$ for the Full Moon, and $f\left(t\right)={\lambda }_M-{\lambda }_S-{270}^{\circ }$ for the last quarter. Because the times are determined from geocentric coordinates, they are independent of location on the Earth [96]. The variation of ${\lambda }_M$ and ${\lambda }_S$ as well as the New Moons are illustrated in Figure 6. This figure also shows the variation of the ecliptic z coordinate $z_{ecl}$ .

Table 1. Mean values of the different orbital periods.

Figure 6. Variation of (a) the geocentric ecliptic longitudes of the Moon, ${\lambda }_M$ , and the Sun, ${\lambda }_S$ , and (b) the ecliptic z coordinate, $z_{ecl}$ , for the Period I, derived from the data provided by JPL’s ephemeris DE430. The red open circles in (a) represent the New Moons, and the black open circles in (b) the passages of the Moon through the ascending node, respectively.

According to astrometric calculations of Kramm et al. [29], the first New Moon of Period I took place on May 24, 2009, 12:09 UT1 and the last one on January 4, 2011, 8:59 UT1. As listed in Table 1, the mean synodic month is about 29.53059 days. However, as illustrated in Figure 7(a), the true lunation deviates from the mean one by up to seven hours during the Period I. For this period, the results differ from those derived by Roncoli [101] using JPL’s ephemeris DE403/LE403 by less 420 s, i.e., within the range of the maximum time step of about 600 s used by Kramm et al. during the integration of Equation (2.3).

Obviously, the reliable determination of the New Moon requires accurate astrometric calculations of the geocentric ecliptic longitudes of the Sun ( ${\lambda }_S$ ) and the Moon ( ${\lambda }_M$ ). It can be achieved the best within the framework of the calculation of well-known geocentric quantities of the Moon.

The geocentric distance, the astrometric right ascension, and the declination of the Moon for the Period I are illustrated in Figure 8. As shown in Figure 8(a), the perigee distance varies significantly more with time than the apogee distance. According to our astrometric calculations, Moon’s perigee distance ranges from 356,593 km to 369,733 km while the apogee distance only ranges from 404,167 km to 406,541 km during the Period I. The corresponding mean geocentric distance of the Moon amounts to 384,792 km.

The actual geocentric distance of the Moon is given by

Figure 7. Difference from (a) the mean synodic month, (b) the mean draconic month, and (c) the mean anomalistic month and the mean synodic month for the Period I, derived from the data provided by JPL’s ephemeris DE430.

Figure 8. Variation of (a) Moon’s geocentric distance, (b) the astrometric right ascension, and (c) the declination of the Moon’s center during the Period I, derived from the data provided by JPL’s ephemeris DE430. The blue open circles in (a) mark the perigee.

$r=+\sqrt{x_{eq}^2+y_{eq}^2+z_{eq}^2}$ (3.9)

Here, $x_{eq}$ , $y_{eq}$ , and $z_{eq}$ are the equatorial coordinates, where $x_{eq}$ points to the first point of Aries that corresponds to the vernal equinox of the Northern Hemisphere. These equatorial coordinates are provided by JPL’s ephemeris DE430.

Perigee and apogee can simply be determined using the first and second derivative tests. In accord with Equation (3.9), the first derivative of the actual geocentric distance with respect to time is given by

$\frac{\text{d}r}{\text{d}t}=\frac{1}{r}\left(x_{eq}\frac{\text{d}{x}_{eq}}{\text{d}t}+y_{eq}\frac{\text{d}{y}_{eq}}{\text{d}t}+z_{eq}\frac{\text{d}{z}_{eq}}{\text{d}t}\right)$ (3.10)

The ephemeris DE430 provides both the components of the position vector and the components of the velocity vector for computing this derivative. The optimum is given when the expression in parentheses is equal to zero (i.e., $\text{d}r/\text{d}t\cong 0$ ). The sign of the second derivative for the time of the optimum marks either the perigee ( ${\text{d}}^{2}r/\text{d}{t}^2>0$ ) or the apogee ( ${\text{d}}^{2}r/\text{d}{t}^2<0$ ). Because ${\text{d}}^2{x}_{eq}/\text{d}{t}^2$ , ${\text{d}}^2{y}_{eq}/\text{d}{t}^2$ , and ${\text{d}}^2{z}_{eq}/\text{d}{t}^2$ are not delivered by the ephemeris, we determine the second derivative numerically from the $\text{d}r/\text{d}t$ curve. The results for about twelve anomalistic months of 2010 are illustrated in Figure 9.

In addition to the sidereal and synodic months, three other orbital periods or months are distinguished (e.g., [94 - 96 , 99]): 1) The orbital period with respect to the equinox called the tropical month, 2) the draconic (or nodical) month defined as the interval between two successive passages of the Moon through the ascending node (see Figure 6(b)), and 3) the anomalistic month related to the interval required by the Moon to move in its path around the Earth from perigee to perigee (see Figure 8(a)).

It is well-known that the lengths of these different months vary with time (e.g., [94 , 100]). For convenience, their mean values are listed in Table 1. The deviations of these different months from their mean values during the Period I are illustrated in Figure 7. While the synodic and draconic months deviate from their mean values only by hours, the anomalistic month departs from its mean even by days. The largest negative deviations occur around the points of inflection in the curve representing the departure of the synodic month from its mean (see Figure 7(c)).

As pointed out by Paige et al. [57], the Moon experiences seasonal insolation variations due to the combined effects of the 5.14˚ obliquity of the Moon’s orbital plane relative to the ecliptic, and the 6.68˚ obliquity of the Moon’s spin axis relative to the Moon’s orbital plane. The net effect is that the latitude of the subsolar point undergoes a seasonal variation with an amplitude of about 1.54˚ and a period of about 346 days, which is less than a full Earth year due to the precession of the Moon’s orbital plane. According to Figure 5(a), the time interval between the two passages of the Sun through the ascending node (characterized by the open circles) of the lunar equator plane corresponds to 347.29 days, i.e., it is slightly longer than the mean draconic year of 346.62 days [57 , 96 , 99] that is based on the 223 synodic months of the Saros period of about 18^a11^d8^h and 19 eclipse years [94 , 99 , 100]. Based on our calculations, these ascending nodes are related to January 11, 2010, 2:24:00 UT1 (TDB = 2455207.60) and December 24, 2010, 9:21:36 UT1 (TDB = 2455554.89). The draconic year is connected with the occurrence of eclipses [99 , 102].

4. MODEL RESULTS

The variations of the solar insolation and the absorbed solar radiation, and the soil heat flux at the depths of 2 cm for numerous parallels of latitude for Moon’s northern and southern hemispheres predicted

Figure 9. First and second derivative tests for determining the perigee ( $dr/dt\cong 0$ and $d^{2}r/d{t}^2>0$ , open circles) and the apogee ( $dr/dt\cong 0$ and $d^{2}r/d{t}^2<0$ , open triangles) for about twelve anomalistic months of 2010. The results are based on the data provided by JPL’s ephemeris DE430.

for the Period II are illustrated in Figure 10. The distributions of the daily mean values of the solar insolation at Moon’s surface and the absorbed solar radiation predicted for this period are illustrated in Figure 1(c) and Figure 1(d), respectively. The variations of the respective slab temperatures are illustrated in Figure 11.

Figure 10. Variations of the solar insolation and the absorbed solar radiation at the surface, and the soil heat flux at the depths of 2 cm for various parallels of latitude for Moon’s northern hemisphere ((a), (c), (e)) and southern hemisphere ((b), (d), (f)) predicted for the Period II.

Since the rotation axis of the Moon is tilted at an angle of about 1.54˚ with respect to the normal of the ecliptic plane, the lunar polar circles are located at latitudes of ±88.46˚ and insolation conditions at these lunar latitudes depend on both local time and season [57]. Figure 10 and Figure 11 already illustrate this dependence, but Figure 12 and Figure 13 exhibit this dependence in a more convenient manner. They illustrate the variations of the solar insolation and the local surface temperature $T_s$ at latitudes of 89˚N, 90˚N, 89˚S and 90˚S for the respective white nights of the polar regions, where $T_s$ is based on a local radiative equilibrium expressed by Equation (1.12). For comparison, the local slab temperature $T_{slab}$ provided by the multilayer-force restore method is illustrated as well. Obviously, the thermal inertia of the system, expressed by the thermal inertial coefficient $R_t$ in Equation (2.3) and the heat flow $H_{sl}$ in the regolith, are responsible for the differences between $T_s$ and $T_{slab}$ . Outside these polar white nights, $T_s$ drops to 0 K during nighttime. Therefore, Equation (1.12) is an unsuitable approximation of Equations (2.3) and (2.5). for most regions of the Moon, as already demonstrated by Wesselink [62].

Figure 11. Variations of the slab temperature for numerous parallels of latitude for Moon’s northern hemisphere (a) and southern hemisphere (b) during Period II (adopted from Kramm et al. [29]).

Figure 12. Variation of (a) the solar insolation and (b) the surface temperature $T_s$ given by Equation (1.12) and the slab temperature $T_{slab}$ according to Equation (2.3) for a white night of the polar region of the northern hemisphere.

Figure 13. As in Figure 12, but for a white night of the polar region of the southern hemisphere.

According to Espenak and Meeus [94 , 100], an annular solar eclipse occurred on January 15, 2010, and a total lunar eclipse occurred on December 21, 2010 (Saros 125). The latter occurred during the twelfth lunation of the Period II. Furthermore, a partial lunar eclipse occurred on June 26, 2010 (Saros 120). To estimate the effect of these two lunar eclipses on the global mean of the surface temperature, we considered the surface temperature measurements performed at the landing site of the Apollo 15 mission, Hadley Rille/Apennine Mountains (26˚5'N, 3˚40'E,) from July 31, 1971 to December 31, 1974 during the Heat Flow Experiment using the Apollo Lunar Surface Experiment Package (ALSEP) [72]. From this dataset, we considered the period from January 8, 1974 (Full Moon) to December 29, 1974 (Full Moon) that is comparable with twelve synodic months. Figure 14 shows the time series TC2 of the probe 2 at the Hadley Rille site, the partial lunar eclipse from June 4, 1974 (Saros 120), and the total lunar eclipse from November 29, 1974 (Saros 125). A comparable drop in the surface temperature of about 220 K during a total lunar eclipse was already observed and modeled by Fontain et al. (1976). From this time series at the Hadley Rille site, we computed the zonal average of the surface temperature by including these two lunar eclipses. We obtained a zonal average of 201.9 K. Then, we repeated our calculation by considering the same dataset in which, however, the effects of these lunar eclipses were arbitrarily removed by using an interpolation procedure for bridging the surface-temperature decreases. Ignoring these lunar eclipses leads to a zonal average of 202.0 K, i.e., without the two lunar eclipses the zonal average would be 0.1 K higher.

5. THE DIVINER LUNAR RADIOMETER EXPERIMENT

The Diviner Lunar Radiometer Experiment (DLRE) is one of seven instruments aboard NASA’s Lunar Reconnaissance Orbiter (LRO) [59 - 61]. The LRO was launched on June 18, 2009 and entered a lunar orbit five days later. Beside the exploration mission of the LRO not considered here, the DLRE was the first experiment to systematically map the global thermal state of the Moon and its diurnal and seasonal variability [57]. Paige et al. [57] described the instrument, its specifications, the spectral channel passbands, and the archived data products in detail. The data are based on a nine-channel radiometer that maps solar reflectance using channels 1 and 2 of high and reduced sensitivity (0.35 - 2.8 μm passband), and infrared emission using four thermal channels, where channel 6 (13 - 23 μm passband) is most sensitive for $T_s>178\text{K}$ , channel 7 (25 - 41 μm passband) is most sensitive for $69\text{K}\le T_s\le 178\text{K}$ , channel 8 (50 - 100 μm passband) is most sensitive for $43\text{K}\le T_s\le 69\text{K}$ , and channel 9 (100 - 400 μm passband) is most sensitive for $T_s<43\text{K}$ . Figure 15 shows the nine spectral passbands of the DLRE. The respective blackbody

Figure 14. (a) Thermocouple temperature TC2 of the probe 2 at the landing site Hadley Rille/Apennine Mountains (26˚5'N, 3˚40'E) of the Apollo 15 for twelve synodic months, (b) partial lunar eclipse from June 4, 1974 (Saros 120), and total lunar eclipse from November 29, 1974 (Saros 125).

Figure 15. The nine spectral passbands of the Diviner Lunar Radiometer Experiment (DLRE) [39].

curves for both the solar range at 1 AU and for various possible surface temperature of the Moon in the infrared range based on Planck’s [103] radiation function are illustrated in Figure 16.

The bolometric temperatures of the DLRE is a measure of the spectrally integrated flux of infrared radiation emerging from the surface [104]. These bolometric temperatures are available since April 2018 under http://luna1.diviner.ucla.edu/~jpierre/diviner/level4_raster_data/ (referenced to Williams et al. [39]). They comprise 24 datasets for various subsolar longitudes ranging from ${\varphi }_{ss}=0^{\circ }\text{E}$ to ${\varphi }_{ss}={345}^{\circ }\text{E}$ , spaced by $\Delta {\varphi }_{ss}={15}^{\circ }$ . Williams et al. [39] compiled all nadir observations (defined to be emission angles <10˚ relative to a sphere) from July 5, 2009 to April 1, 2015 (over 25,000 orbits) into bins of 0.5˚ latitude and longitude and 0.25 h of local time. Because these authors discussed the accuracy of the DLRE observations in detail, the reader is referred to their paper and the cited sources, especially [57 , 104 - 107].

Figure 16. Representative blackbody curves for the solar range at 1 AU and for various possible surface temperature of the Moon in the infrared range that are based on Planck’s [103] blackbody radiation function (adopted from Paige et al. [57], but with the solar spectral emittance at 1 AU added).

6. MODEL RESULTS VERSUS DLRE OBSERVATIONS

Figure 17 and Figure 18 illustrate the results of the comparison between $T_{slab}$ of the eleventh lunation shown in Figure 11 (which is close in length to the mean synodic month, see Figure 7(a)) and $T_{bol}$ of the DLRE related to the subsolar longitude ${\varphi }_{ss}={180}^{\circ }\text{E}$ [39] for the equator and various parallels of latitude of the northern hemisphere and the southern hemisphere, respectively. Obviously, the predicted slab temperatures follow the bolometric temperatures acceptably. Between 75˚N and 75˚S, (most important to compute $\langle T_{slab}\rangle$ and $\langle T_{bol}\rangle$ , respectively, see Figure 19), the zonally averaged differences and the respective standard deviations are: 3.5 K ± 12.0 K for 75˚N 1.4 K ± 8.0 K for 60˚N, 0.9 K ± 6.5 K for 45˚N, 0.7 K ± 6.4 K for 30˚N, 0.4 K ± 6.3 K for 15˚N, 0.3 K ± 5.8 K for 0˚, 0.0 K ± 5.1 K for 15˚S, 0.1 K ± 5.8 K for 30˚S, 0.4 K ± 6.3 K for 45˚S, −0.4 K ± 8.1 K for 60˚S, and 0.2 K ± 10.2 K for 75˚S.

Figure 19(a) shows meridional distributions of the zonal averages of the bolometric temperature, ${\bar{T}}_{bol}\left(\theta \right)$ , for various zenith angles $\theta$ that are based on Equation (1.3) choosing $\psi \left(\theta ,\phi \right)=T_{bol}\left(\theta ,\phi \right)$ . These meridional distributions represent the range of all 24 DLRE datasets. Figure 19(b) shows the corresponding meridional distributions of ${\bar{T}}_{bol}\left(\theta \right)\sin \theta$ , as required by Equation (1.2) for global averaging. Obviously, the higher nighttime temperatures and standard deviations for the polar spherical caps at latitudes beyond ±80˚ (i.e., low and high values of $\theta$ ) resulting from the occurrence of low-angle illumination of surfaces, especially during polar summers, as already reported by Williams et al. [39], are of minor importance in global averaging. Figure 19 also shows the slab-temperature results of Kramm et al. [29] provided by their multilayer force-restore method. These results agree the best with the distribution of the bolometric temperature related to ${\varphi }_{ss}={180}^{\circ }\text{E}$ . In this case, $\langle T_{slab}\rangle =197.9\text{K}$ and $\langle T_{bol}\rangle =200.0\text{K}$ .

The zonal averages derived from the 24 DLRE datasets and the respective standard deviation for various parallels of latitude are listed in Table 2. Also listed are the zonal averages obtained from model simulations using the multilayer-force-restore method [29], and the in-situ measurement of the Apollo 15 and 17 missions using the probe 2 of the Heat Flow Experiment (HFE) of the Apollo Lunar Surface Experiment Package (ALSEP) [72 , 73], where the entire periods were considered. Thus, these results slightly

Figure 17. Comparison between the slab temperature, $T_{slab}$ , of the eleventh lunation shown in Figure 11 (adopted from Kramm et al. [29]) and the bolometric temperature, $T_{bol}$ , of the Diviner Lunar Radiometer Experiment related to the subsolar longitude ${\varphi }_{ss}=180˚E$ (with reference to Williams et al. [39]) for the Equator and various parallels of latitude of the northern hemisphere. Note that the eleventh lunation is close in length to the mean synodic month (see Figure 7(a)). Furthermore, the $T_{bol}$ curves are averages for the respective parallels of latitude.

Figure 18. As in Figure 17, but only for the southern hemisphere.

Figure 19. (a) Typical meridional distributions of the zonal averages (see Equation (1.3)) of the bolometric temperature, ${\bar{T}}_{bol}\left(\theta \right)$ , for various values of the zenith angle $\theta$ and the subsolar longitude ${\varphi }_{ss}$ , and (b) the corresponding meridional distributions of ${\bar{T}}_{bol}\left(\theta \right)sin\theta$ as required by Equation (1.2) for global averaging. The black open circles in (a) represent the zonal averages listed in Table 2 that were derived from the 24 datasets of the Diviner Lunar Radiometer Experiment [39].

differ from those published by Keihm et al. [72] and Keihm and Langseth [73].

The global averages of the bolometric temperature, $\langle T_{bol}\rangle$ , for all 24 datasets related to the subsolar longitude ${\varphi }_{ss}$ are illustrated in Figure 20. The arithmetic mean and the standard deviation of this

distribution amount to $\langle T_{bol}\rangle =201.1\text{K}\pm 0.6\text{K}$ . Since ${\langle T_{bol}^4\rangle }^{1/4}=271.0\text{K}\pm 0.7\text{K}$ , averaged over these 24 datasets, we have $\langle T_{bol}\rangle \cong 0.742{\langle T_{bol}^4\rangle }^{1/4}$ . This means that the DLRE observations confirm the result $\langle T_{slab}\rangle \cong 0.743{\langle T_{slab}^4\rangle }^{1/4}$ theoretically derived by Kramm et al. [29]. These observations also confirm the

temperature inequality (1.17) of Gerlich and Tscheuschner [53].

The meridional distributions of the zonal averages of the absorbed solar radiation, $\bar{Q}\left(\theta \right)$ , and the emitted infrared radiation, ${\bar{F}}_{IR}\left(\theta \right)$ , both weighted by $\sin \theta$ as required for global averaging, are shown in Figure 21 for different values of the normal albedo and the integral relative emissivity, ${\alpha }_0=0.08$ and $\epsilon =0.95$ derived by Williams et al. [39] from the DLRE observations, and ${\alpha }_0=0.10$ and $\epsilon =0.98$ suggested by Vasavada et al. [58] and used by Kramm et al. [29] in their model simulations. It seems that in the case of ${\varphi }_{ss}={180}^{\circ }\text{E}$ , the normal ${\alpha }_0=0.08$ is a little bit too low. In this case, the global average of the absorbed solar radiation computed for the twelve lunations analyzed by Kramm et al. is $\langle Q\rangle =288.7\text{W}\cdot {\text{m}}^{-2}$ for ${\alpha }_0=0.08$ . Whereas the global average of the emitted infrared radiation is $\langle F_{IR}\rangle =285.7\text{W}\cdot {\text{m}}^{-2}$ for $\epsilon =0.95$ leading to a radiative imbalance of $\langle Q\rangle -\langle F_{IR}\rangle =3.0\text{W}\cdot {\text{m}}^{-2}$ . The predicted results of Kramm et al. provided by the multilayer-force-restore method read $\langle Q\rangle =279.9\text{W}\cdot {\text{m}}^{-2}$ for ${\alpha }_0=0.10$ and $\langle F_{IR}\rangle =280.0\text{W}\cdot {\text{m}}^{-2}$ for $\epsilon =0.98$ leading to a radiative imbalance of −0.1 W·m⁻².

Based on the globally averaged emitted infrared radiation of $\langle F_{IR}\rangle =290.5\text{W}\cdot {\text{m}}^{-2}\pm 3.0\text{W}\cdot {\text{m}}^{-2}$ derived from these 24 datasets (see Figure 22), the effective radiative temperature for the Moon would be

Table 2. Zonal averages for various parallels of latitude obtained from model simulations using the multilayer-force-restore method ( ${\bar{T}}_{slab}$ ) [29], the Diviner Lunar Radiometer Experiment ( ${\bar{T}}_{bol}$ ) [39], and the in-situ measurement of the Apollo 15 and 17 missions using the probe 2 data of the Heat Flow Experiment (HFE) of the Apollo Lunar Surface Experiment Package (ALSEP) [72 , 73].

^*For the latitude 20.25˚N. ^#For the latitude 26.25˚N.

Figure 20. Globally averaged bolometric temperature, $\langle T_{bol}\rangle$ , for all 24 DLRE datasets related to the subsolar longitude ${\varphi }_{ss}$ .

Figure 21. Meridional distributions of the zonal averages of the absorbed solar radiation, $\bar{Q}\left(\theta \right)$ , and the emitted infrared radiation, ${\bar{F}}_{IR}\left(\theta \right)$ , both weighted by $sin\theta$ as required by Equation (1.2) for global averaging.

$T_{e,M}={\langle T_{bol}^4\rangle }^{1/4}=271.0\text{K}\pm 0.7\text{K}$ . Figure 22 also shows the emission of infrared radiation calculated by

applying the Stefan-Boltzmann power law to the globally averaged bolometric temperatures illustrated in Figure 20, i.e.,

$\epsilon \sigma {\langle T_{bol}\rangle }^4<\langle \epsilon \sigma T_{bol}^4\rangle$ (6.1)

Figure 22. Globally averaged emitted infrared radiation $\langle \epsilon \sigma T_{bol}^4\rangle$ , derived from the 24 DLRE datasets of the bolometric temperature related to the subsolar longitude ${\varphi }_{ss}$ . Also illustrated are the results provided by the inadequate application of the Stefan-Boltzmann law to the global mean of the bolometric temperature expressed by $\epsilon \sigma {\langle T_{bol}\rangle }^4$ .

Figure 23. Spectral distribution of solar radiation at the TOA, and a typical distribution of that which reaches the Earth’s surface, compared with the radiation emitted by a blackbody at a temperature of 6000 K (adopted from Coulson [110]).

Apparently, this kind of calculation is meritless. Therefore, it is time to acknowledge that the Stefan-Boltzmann power law must not be applied to globally averaged temperatures.

The DLRE data provide observational evidence that the concept of the effective radiation temperature must be discarded for planets and their natural satellites. In the case of stars, for which the concept of the effective radiation temperature was derived, the condition of a uniformly distributed emittance may be crudely fulfilled, and the stars’ emission spectra may be approximated by Planck’s blackbody radiation function related to their effective radiation temperatures, as illustrated in Figure 23 for our Sun. The condition of a uniform distribution of the emittance, however is, by far, not fulfilled in the case of planets and their natural satellites. In addition, their emission spectra vary with latitude and time. This fact is the reason why radiometers use different channels as illustrated by Figure 15 and Figure 16 to cover the wide range of surface temperatures. Since the maximum of the intensity of Planck’s blackbody radiation function is proportional to the fifth power of the temperature [8 , 86 , 108 , 109], averaging over the Planck functions for different temperatures is physically and mathematically awkward.

7. SUMMARY AND CONCLUSIONS

In our paper, we discussed the solar climate of the Moon and the distribution of its resulting surface temperature. Since the solar climate, when handled in its historical sense, would lead to a surface temperature of 0 K for the dark site of the Moon, it is essential to expand the concept of solar climate in such a way that such an artifact is generally excluded. Consequently, the local radiation balance (Equation (1.12)) must be replaced, at least, by a local energy flux budget (Equation (2.5)) to compute the related distribution of the surface temperature in a realistic manner. In this study, we used the multilayer-force-restore method by Kramm et al. [29] which applies the more advanced local energy budget expressed by Equation (2.3) to compute the distribution of the slab temperature, $T_{slab}$ of the Moon. Predicted slab temperature is considered as the realistic surface temperature, and in a further step the global average of Moon’s surface temperature. Their astrometric results derived from the data provided by JPL’s planetary and lunar ephemeris DE430 underlined that the calculated solar insolation, required by their multilayer-force-restore method [29], meets the required accuracy criteria.

The numerical prediction was performed for 20 lunations starting May 24, 2009, 00:00 UT1 (Period I). Based on the astrometric results, we found for this period a mean heliocentric distance for the Moon of 1.00124279 AU and for the Earth of 1.00166376 AU. The corresponding mean geocentric distance of the Moon amounts to 384,792 km. Furthermore, we found that the synodic months deviate from their mean in a range of −5.7 h to 6.9 h and the draconic months depart from their mean by ±3.4 h. Whereas the anomalistic months depart from their mean in a range of −2.83 d to 0.97 d with the largest negative deviations occurring around the points of inflection in the curve that represents the departure of the synodic month from its mean. Moreover, based on the two successive passages of the Sun through the ascending node of the lunar equator plane during that Period I, we found that the time interval between them corresponds to 347.29 days, i.e., it is slightly longer than the mean draconic year of 346.62 days.

Because the multilayer-force-restore method has to be spun up to equilibrium prior to analysis of the results, we analyzed only the results provided by it for the last 12 lunations starting January 15, 2010, 07:11 UT1 (Period II). As shown in our paper, the predicted slab temperatures follow the bolometric temperatures in an acceptable manner. Between 75˚N and 75˚S, most important to compute $\langle T_{slab}\rangle$ and $\langle T_{bol}\rangle$ , the zonally averaged differences and the respective standard deviations are: 3.5 K ± 12.0 K for 75˚N 1.4 K ± 8.0 K for 60˚N, 0.9 K ± 6.5 K for 45˚N, 0.7 K ± 6.4 K for 30˚N, 0.4 K ± 6.3 K for 15˚N, 0.3 K ± 5.8 K for 0˚, 0.0 K ± 5.1 K for 15˚S, 0.1 K ± 5.8 K for 30˚S, 0.4 K ± 6.3 K for 45˚S, −0.4 K ± 8.1 K for 60˚S, and 0.2 K ± 10.2 K for 75˚S. Based on the observations performed at the landing site of the Apollo 15 mission, we showed that the occurrence of the two lunar eclipses has only a negligible effect on the zonal averages of the surface temperature if twelve lunations are considered.

The global averages of the bolometric temperature, $\langle T_{bol}\rangle$ for all 24 DLRE datasets related to the subsolar longitude ${\varphi }_{ss}$ amount to $\langle T_{bol}\rangle =201.1\text{K}\pm 0.6\text{K}$ . Based on the globally averaged emitted infrared radiation of $\langle F_{IR}\rangle =290.5\text{W}\cdot {\text{m}}^{-2}\pm 3.0\text{W}\cdot {\text{m}}^{-2}$ derived from these 24 DLRE datasets, the effective

radiative temperature of the Moon is $T_{e,M}={\langle T_{bol}^4\rangle }^{1/4}=271.0\text{K}\pm 0.7\text{K}$ so that $\langle T_{bol}\rangle \cong 0.742T_{e,M}$ . This

means that in the case of the Moon, the effective radiation temperature is about 60 K higher than the globally averaged surface temperature.

Furthermore, our results obtained by means of the DLRE observations confirm Kramm et al. [29]

who obtained for the Moon $\langle T_{slab}\rangle \cong 197.9\text{K}$ and $T_{e,M}={\langle T_{slab}^4\left(\theta ,\phi \right)\rangle }^{1/4}\cong 266.4\text{K}$ resulting in

$\langle T_{slab}\rangle \cong 0.743T_{e,M}$ . Our results also empirically confirm the temperature inequality (1.17) of Gerlich and

Tscheuschner [53], i.e., $\langle T_{bol}\rangle <{\langle T_{bol}^4\rangle }^{1/4}=T_{e,M}$ .

Obviously, the relationship between the global average of the surface temperature and the effective radiative temperature of a rocky celestial body differs by a factor that depends on the astronomical parameters, especially on the angular velocity of rotation. Kramm et al. [29], for instance, found for the solar climate of the 27.4 times faster rotating Earth $\langle T_{slab}\rangle \cong 0.828T_{e,E}$ . Consequently, the DLRE observations provide empirical evidence that in the case of rocky planets and their natural satellites, the globally averaged surface temperature is notably lower than their effective radiation temperature. Finally, based on the 24 DLRE datasets our study showed that applying the Stefan-Boltzmann power law to the globally averaged bolometric temperatures provides meritless results for rocky celestial bodies.

ACKNOWLEDGEMENTS

We would like to express our thanks to the JPL team around Drs. James G. Williams and William M. Folkner for making the planetary and lunar ephemeris DE 430 and the respective FORTRAN subroutines asc2eph, testeph, and PLEPH available for us, and to the anonymous reviewers for helpful comments and suggestions.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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