A Real-Time Model of the Seasonal Temperature of Lunar Polar Region and Data Validation

1892 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 58, NO. 3, MARCH 2020 A Real-Time Model of the Seasonal Temperature of Lunar Polar Region and Data Validation

Niutao Liu , Student Member, IEEE,andYa-QiuJin , Life Fellow, IEEE

Abstract— A small tilt in the spin axis of the moon over the properties of the regolith media have been well studied for data ecliptic plane causes seasonal incidence variation of solar illumi- validation [8], [9]. Temperature maps of lunar polar in summer nation and, especially, causes significant temperature difference and winter are generated with the Diviner data collected before at the polar region. In this article, following the position of the subsolar point, the real-time model of solar illumination the end of 2017 [10]. incidence over the moon polar region is developed. Based on Then, in 2007 and 2010, respectively, Chinese this model with solving the 1-D heat conduction equation, Chang’e-1 and Chang’e-2 (CE-1,-2) with four-channel the seasonal temperature of the lunar surface is obtained and is in microwave (MW) radiometers made the first measurement of agreement with the Diviner infrared (IR) data. Meanwhile, using MW brightness temperature (TB) of the lunar regolith. Due to the fluctuation-dissipation theorem and the Wentzel–Kramers– Brillouin (WKB) approach for lunar regolith media, the seasonal the MW penetration depth, this TB contains contribution from microwave (MW) brightness temperature (TB) is also obtained the regolith media at the depth as deep as 3–5 m [11], [12]. and validated by Chang’e-2 (CE-2) 37-GHz TB data. These data CE-2 TB at 37 GHz has been employed in the fusion study also show that the lunar surface temperature and the MW TB of the IR surface temperature [13]–[15]. even in the permanent shaded region (PSR) undergo seasonal Due to cosine variation, the temperature and TB around variation as well. It might be due to the seasonal thermal radiation on the topographic PSR coming from the sunlit crater the lunar polar are especially sensitive to the incidence angle walls caused by seasonal temperature variation. The Diviner IR of solar illumination. In other words, a small change in the data show that the highest temperature in the Hermite-A crater incident angle causes large temperature variation. The position at the north polar PSR can reach 109 K in summer. of the subsolar point can be determined by the planetary theory Index Terms— Chang’E (CE), infrared (IR) and microwave VSOP82 and the Chapront ELP-2000/82 lunar theory, which (MW) brightness temperature (TB), moon, permanent shaded includes the 18.6-year nodal precession [16]. Then, the real- region (PSR), seasonal temperature variation. time model of the incident angle of solar illumination can I. INTRODUCTION be developed. Solving the 1-D heat conduction equation with the thermal–physical profiles, the seasonal temperature of the HE lunar polar regions are always of low temperatures lunar polar region for a year period can be calculated. due to less illumination at high latitude. Especially, T In this article, the Diviner T channel (50–100 μm) is the permanent shaded region (PSR) totally without solar 8 employed. Calculation is then well validated by the Diviner illumination is in extremely low physical temperature [1]. ◦ IR data. Meanwhile, using the fluctuation-dissipation theorem A small tilted angle, 1 32 , of the spin axis of the moon over and the WKB approach for the lunar regolith media with a the ecliptic plane yields different incidence angles of the solar temperature profile [14], [15], the MW TB at 37 GHz for a illumination during a year’s seasons [2], [3]. To take account of year period can be calculated and validated by CE-2 data as the illumination condition, the topography of the lunar polar well. Especially, by checking these data in the PSR, it is found surface had been also studied [4]–[6]. Moreover, the small that there is also seasonal variation of IR temperature and MW tilted angle would also cause seasonal temperature variation TB in the PSR even without solar illumination. The seasonal of the lunar surface. variation in the MW TB was studied with CE-2 data [17]. As one of the most important lunar remote sensing This article is organized as follows. In Section II, the motion programs, the Diviner onboard the LRO measured the of the moon is briefly illustrated. Section III presents the real- mid-infrared (IR) irradiance of the lunar surface [7]. Based time model of the lunar surface temperature and the simulation on the Diviner IR data, the parameters of the thermal–physical of the seasonal variation of the temperature during a year and Manuscript received April 18, 2019; revised July 2, 2019 and the validation by the Diviner IR data. The seasonal and diurnal October 16, 2019; accepted October 24, 2019. Date of publication temperature variations, especially, in the PSR are discussed as November 19, 2019; date of current version February 26, 2020. This work was supported in part by the National Key Research and Development Program of well. In Section IV, the seasonal variation of the MW TB is China under Grant 2017YFB0502703. (Corresponding author: Ya-Qiu Jin.) simulated and validated by the CE-2 37-GHz TB data. The N. Liu is with the Key Laboratory of EMW Information, Fudan University, seasonal MW TB in the PSR is also found. Finally, Section V Shanghai 200433, China. Y.-Q. Jin is with the Key Laboratory of Wave Scattering and Remote gives the conclusion. Sensing Information, Fudan University, Shanghai 200433, China (e-mail: [email protected]). II. MOTION OF THE MOON Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. In 1683, Cassini stated three empirical laws: 1) the moon Digital Object Identifier 10.1109/TGRS.2019.2950300 rotates eastward about a fixed axis with a constant angular 0196-2892 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information.

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Fig. 2. Global and local coordinate systems.

when the subsolar point is at the south hemisphere, it is deﬁned as the summer in the southern hemisphere and the winter in the northern hemisphere, and vice versa [Fig. 1(b)].

III. REAL-TIME MODEL OF LUNAR SURFACE TEMPERATURE A. Real-Time Information Based on the position of the subsolar point, a real-time Fig. 1. Position of the subsolar point. (a) Subsolar longitude. (b) Subsolar model is presented. The real-time total solar irradiance (TSI) latitude. A positive latitude is at the northern hemisphere and a negative and illumination incidence angle are used as the inputs of latitude is at the southern hemisphere. The season depends on the subsolar latitude. the heat conduction model to obtain the real-time temperature profile of the lunar regolith. / 2 velocity and in a period equal to that for one complete TSI is given as 1371W m when the distance between the revolution about the earth; 2) the planes of the moon’s equator sun and the moon is 1 Astronomical Unit (AU) [19]. The ± . and the earth’s orbit (ecliptic) meet at a fixed angle, namely, distance between the earth and the sun is about 1 0 02AU, 1◦32; and 3) the normal to the ecliptic, the normal to the plane and the distance between the moon and the earth is about of the moon’s orbit, and the axis of rotation of the moon all 0.0025 AU. Here, the distance between the moon and the lie on the same plane [18]. Due to the tilt in the spin axis, earth is not considered. The real-time TSI is determined by the incidence angle of solar illumination varies at different the distance between the earth and the sun as θ seasons. The solar incidence angle i is very large toward TSI = 1371/d2 (1) the polar region. The solar irradiance on the unit area can se be expressed as TSI × cos θi , where TSI is the total solar where dse is the distance between the earth and the sun in the irradiance on the moon [19]. Due to the functional dependence unit of AU. At the aphelion and perihelion, TSI is 1326W/m2 of cosine, for example, given the incidence angle on a flat and 1418W/m2, respectively. ◦ ◦ region changing from 88.5 into 87 , the solar irradiance on θi is the incident angle of solar illumination on the lunar this unit lunar surface doubles. This small angular change in surface, which is determined by the position of the subsolar the polar region can cause huge temperature variation. It takes point, the region location, and its surface topography, i.e., the ascending node of the moon about 18.6 years to move surface slope denoted by the angles (ω, γ ). Suppose that the through 360◦ relative to the vernal equinox, and the direction subsolar longitude is λ and the subsolar latitude is ϕ.The of motion is westward [20]. The period of the axis precession longitude of the local region is λ0 and the latitude is ϕ0.The is 18.6 years as well, resulting in the periodic change of moon is seen as an ideal sphere with the radius of 1737.4 km. Lunar’s seasons. Fig. 2 shows the orthogonal coordinate system, where zˆ is the The planetary theory VSOP82 and the Chapront ELP-2000/ unit vector from the moon center to the north pole, and xˆ is the 82 lunar theory are adopted to obtain the position of the unit vector from the moon center to the lunar surface center subsolar point [16]. VSOP82 consists of long series of periodic (both the longitude and latitude are 0◦). terms, which can be used to obtain the orbital parameters, such At the region point, its local coordinate system is estab- as the semimajor axis of the orbit, for each of the major planets lished: the axis xˆl is parallel to the latitude line of the region from Mercury to Neptune. The periodic terms in [21] are used point in the anticlockwise direction seen from the north pole. to obtain the subsolar point, which includes the 18.6-year The axis yˆl is parallel to the longitude line of the region point nodal precession. The subsolar longitude and latitude from in the direction of the increasing latitude from 90◦ Sto90◦ N. 2010 to 2012 are shown in Fig. 1(a) and (b). In this article, The axis zˆl is the unit vector from the moon center to the

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region point. xˆl , yˆl,andzˆl can be expressed as where =− . −1 −1, =+ . −1 −2 xˆl =−sin λ0 ×ˆx + cos λ0 ×ˆy (2a) C0 3 6125J kg K C1 2 7431J kg K =+ . × −3 −1 −3 yˆl =−sin ϕ0 × cos λ0 ×ˆx − sin ϕ0 × sin λ0 ×ˆy+cos ϕ0 ×ˆz C2 2 3616 10 Jkg K =− . × −5 −1 −4 (2b) C3 1 2340 10 Jkg K =+ . × −9 −1 −5. zˆl = cos ϕ0 × cos λ0 ×ˆx + cos ϕ0 × sin λ0 ×ˆy + sin ϕ0 ×ˆz. C4 8 9093 10 Jkg K (2c) The heat conductivity K (T ) of the regolith is as follows [8]: The normal vector of the lunar surface is written as 3 −α ·ˆ − β ×ˆ +ˆ T xl yl zl K = Kc 1 + χ . (11) nˆl = (3) 350 α2 + β2 + 1 It consists of solid phonon conductivity and radiative con- where α and β are the local slopes in the xˆl-andyˆl-directions ductivity. The radiative conductivity parameter χ is 2.7 [8]. Kc ∂ ( , ) is the solid phonon of thermal conductivity and is assumed to α = f xl yl (4a) be linearly proportional to the density based on the experi- ∂xl mental data in [25] ∂ f (xl, yl ) β = . (4b) ρ − ρ ∂y d l Kc = Kd − (Kd − Ks ) (12) ρd − ρs f (xl, yl ) is the elevation at the location (xl, yl ). The local = . × −4 −1 −1 = . × slope angles, γ and ω, are used to express α and β where Ks 7 4 10 Wm K and Kd 3 4 10−3 Wm−1 K−1 are the heat conductivities at the surface α = tan(γ ) (5a) and depth, respectively. β = (ω). The boundary conditions of (8) are written as tan (5b) ∂ T + 4 The unit vector from the moon center to the subsolar point K (z, T ) = TSI(1 − A) cos θi − eσ T − J0 (13a) ∂z s can be written as z=0 ∂T K (z, T ) =−J (13b) sˆ = cos ϕ × cos λ ×ˆx + cos ϕ × sin λ ×ˆy + sin ϕ ×ˆz. (6) ∂ 0 z z=−∞ The local incidence angle of solar illumination can be where the real-time TSI is obtained from (1). θi is the incident ˆ ·ˆ< + θ obtained as angle of sunlight. At night, there is zl s 0andcos i is equal to 0. The real-time local incidence angle at daytime can θ =ˆ ×ˆ. cos i nl s (7) be obtained in (7). If cos θi < 0, the lunar surface is shady and + + cos θi is equal to 0. Otherwise, cos θi is equal to cos θi .The The real-time TSI and the incidence angle are used as the bolometric IR emissivity of the lunar surface, e,issettobe input parameters of the heat conduction equation to calculate 0.95 [26]. Ts is the surface temperature. J0 is a constant heat the real-time temperature profile of the lunar regolith media. flux originating from internal heat source and is ignorable [22]. Here, it is set to 0W/m2. σ is the Stefan–Boltzmann constant . × −8 −2 −4) B. Heat Conduction Model (5 67 10 Wm K . The lunar surface albedo A depends on the solar incident The 1-D heat conduction equation is applied to solving the angle θi and is empirically formulated as follows [27]: physical temperature of the regolith media [22] 3 8 A(θ) = A0 + a(θi /45) + b(θi /90) (14) ∂T ∂ ∂T ρC = K (8) ∂t ∂z ∂z where A0 is the normalized albedo and is set to be 0.12, which is in the range of the albedo value derived from where T (z) is the temperature profile at the regolith depth z, Clementine 750-nm data. Parameters a and b are two empir- ρ(z) is the bulk density profile, C is the heat capacity, and K ical coefficients, estimated as 0.06 and 0.25 [8]. At the ◦ is the thermal conductivity. polar region, θi is close to 90 and A(θ) is much larger ρ( ) The density profile z of the regolith is derived based on than A0. the Diviner data as [8] The position information of the subsolar point from Janu- − / ary 1, 2000 to December 31, 2011 is applied to calculate the ρ(z) = ρ − (ρ − ρ )e z H (9) d d s real-time TSI and incidence angle. The FDTD code is running −3 −3 for more than 100 lunations (29.53 earth day) to guarantee the where ρs = 1100 kgm , ρd = 1800 kgm ,andH is set to solution convergence. Provided that the longitude of the local 6cm[9]. ◦ ◦ The heat capacity C(T ) of the regolith was fitted by the region is 21 W and the latitude is 85.1 S, the simulated ω γ measurement of the samples [23], [24] surface temperatures of this region for different slopes , in 2010 are shown in Fig. 3(a) and (b). A slope angle ω 2 3 4 C = C0 + C1T + C2T + C3T + C4T (10) toward the equator may have much direct illumination [19].

the surface toward illumination increases the temperature in both winter and summer, and yields smaller T . The slope angle γ , as given in (5a), changes the local incident angle, and thus inﬂuences the temperature. The minimum incident angle is 77.34◦ when γ =±10◦ and 81.46◦ when γ =±5◦. ItisshowninFig.3(b)thattheγ angle also changes the time when the temperature reaches the peak. A negative γ at the south polar will make the peak time ahead of noon, while a positive will make the peak time after the noon. It also can be seen that the differences in the temperature between the slopes are more signiﬁcant at daytime (peaks) than those at night (bottoms).

C. Seasonal Variation in Diviner Data The Diviner Lunar Radio Experiment onboard LRO has been acquiring solar reflectance and mid-IR radiance measurements since July 2009 [28]. The LRO orbit plane is inclined about 90◦ from the moon equator. The orbit plane is nearly fixed in the inertial space. It costs the moon 27.3 day to rotate 360◦ relative to the LRO orbit plane. This is the length of one Diviner mapping cycle in the level 2 Global Data Records (GDRs) [29]. The data are available at the NASA Planetary Geosciences Node (LRO-L-DLRE-5-GDR- ◦ ◦ V1.0). The map projection type of the level 2 GDR data at Fig. 3. (a) Surface temperature with α = 0 in 2010 at 85.1 S, 21 W. ◦ ◦ (b) Surface temperature with β = 0 in 2010 at 85.1 S, 21 W. the polar region is polar-stereographic. The spatial resolution of the map is 240 m/pixel. The IR TB data used here are in TABLE I the T8 channel (50–100 μm). SIMULATED SEASONAL TEMPERATURE As usual, the IR emissivity of the lunar surface is taken as 1, and its IR TB is equivalent to the surface temperature [9]. The data collected from March 16, 2010 to April 12, 2010 (dgdr_tb8_avg_pols_20100316d_240_img) and from September 10, 2010 to October 7, 2010 (dgdr_tb8_avg_pols_20100910d _240_img) at the south pole are used to illustrate the seasonal temperature change. It is in the winter at the south pole in March and April 2010, because the subsolar latitude is nearby 1.5◦ N, while it is in summer in September and October 2010 as the subsolar latitude is nearby 1.5◦ S. The temperature maps are shown in Fig. 4(a) and (b). The strips in Fig. 4(a) and (b) are the places of no data. To min- For example, a slope of 5◦ may experience equivalent peak imize the effect of local time on the temperature value, three radiance to a flat surface 5◦ closer to the equator. regions of similar local time are picked, as outlined by black The subsolar latitude in March and April in 2010 is nearby boxes in Fig. 4(a). The data were observed during daytime. 1.4◦ N. Thus, it is the winter in the southern hemisphere. While The local time of these three regions is about 12:00. The in September and October in 2010, the subsolar latitude is differences between the local times of the most picked areas about 1.4◦ S, and it is the summer in the southern hemisphere. are less than 0.1 h. Thus, it is assumed that the temperature On this region (21◦ W, 85.1◦ S), the simulated temperature difference caused by local time difference at these areas can difference T between the temperatures in the summer and be ignored. in the winter with different slope angles ω and γ is shown Region 1 locates nearby (87◦ S, 30◦ W), and the local in Table I. time is about 12:15. The temperature difference is obtained The slope angle ω herewith is denoted as negative, because by subtracting the temperature in winter from the temperature a positive ω at the south pole may result in a shaded slope in in summer. the winter. A large positive ω will lead to a PSR. Fig. 5(a)–(d) shows, respectively, the temperature, the tem- Fig. 3 shows that the seasonal temperature difference T perature difference, and the local time difference. The high can reach 39 K with no slope ω = 0. From Table I, it can temperature is caused by the direct solar illumination. The be seen that as the slope changes, e.g., ω from 0 to −20◦, temperature difference at the illuminated slope ranges from 0

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Fig. 6. Temperature and local time of Region 2 in Fig. 4. The horizontal and vertical axes are pixel number. (a) Temperature in summer. (b) Temperature in winter. (c) Temperature difference between (a) and (b). (d) Local time difference.

Fig. 4. Diviner data in (a) summer (dgdr_tb8_avg_pols_ 20100910d_240_img) and (b) winter (dgdr_tb8_avg_pols_ 20100316d_ 240_img). The horizontal and vertical axes denote the pixel number. The regions in the red boxes are used to illustrate the seasonal temperature difference.

Fig. 7. Temperature and local time of Region 3 in Fig. 4. The horizontal and vertical axes are the pixel number. (a) Temperature in summer. (b) Tem- perature in winter. (c) Temperature difference between (a) and (b). (d) Local time difference.

larger than 40 K, can be found in those places that is shaded in winter and exposed to illumination in summer. Figs. 6(a)–(d) and 7(a)–(d), respectively, show the tem- Fig. 5. Temperature and local time of Region 1 in Fig. 4. The horizontal and vertical axes are pixel number. (a) Temperature in summer. (b) Temperature perature in summer and winter, the temperature differ- ◦ in winter. (c) Temperature difference between (a) and (b). (d) Local time ence, and the local time difference of Region 2 (85.1 S, difference. 21◦ W)andRegion3(84.4◦ S, 161.4◦ W). The seasonal temperature variations can be found in comparison with to 40 K. The high temperature at the illuminated slope usually Figs. 6(a)–(d) and 7(a)–(d). corresponds to a small temperature difference, in agreement To see the effects of the local surface slopes on the with the simulation. Huge temperature differences, e.g., much temperature, a part of the temperature data in Fig. 4 and its

Fig. 9. Relationship between the temperature in winter and the slope angles in Fig. 8. (a) γ angle. (b) ω angle.

Fig. 8. Temperature (dgdr_tb8_avg_pols_20100910d_240_img and Fig. 10. Relationship between the temperature difference T (subtract the dgdr_tb8_ avg_pols_20100316d_240_img), DEM (ldem_ 75s_240m_ ﬂoat), temperature in winter from that in summer) and slope angle ω. and the slope angles resampled into cylindrical projection. The number in the box corresponds to the part number in Fig. 10.

is shaded by the crater walls or mountains, it would remain DEM data are resampled into cylindrical projection, as shown low temperature. in Fig. 8(a)–(f), where the horizontal ordinate is the longitude The surface temperature at daytime is determined by the in the range of 3◦ Wto65◦ W, and the vertical ordinate is the solar illumination because of the heat balance [19]. Therefore, latitude in the range of 82◦ Sto88◦ S. One degree in latitude the surface temperature is determined by the local time and the spans 1737.4π/180 km in distance, while 1◦ in longitude slope angles, i.e., ω,γ , which determines the local incidence spans 1737.4π · cos(ϕ0)/180 km in distance. It is easy to see angle. The local time of the data in Fig. 9 is about 12:00. from Fig. 8(a), (b), (d), and (f) that the temperature of the As seen from Fig. 9(b), the temperature is mainly related to illuminated slope is higher that the shady slope. In comparison, the slope angle ω. the slope angle ω seems to have a more signiﬁcant impact on Fig. 10 shows the relationship between the temperature the temperature and the temperature difference at the noontime difference T and the slope angle, ω in Fig. 8. The local time than the γ . differences of the data in Fig. 10 are all less than 0.2 h. The Fig. 9(a) and (b) shows a direct comparison between the number in the box of Fig. 8 corresponds to the part number temperature in winter and the slope angles in Fig. 8. in Fig. 10. From Fig. 9(b), it can be seen that a decrease in ω will In Part 1 ( T < 40 K, ω<0◦), a negative ω at the lead to an increase in temperature. When ω is about −20◦, latitude lower than 88◦ S can make sure the slope not be the highest temperature may approach to 300 K. It is in shaded by itself. The T increases with ω increasing on the agreement with the simulation. If a surface facing the equator illuminated slopes, which is consistent with the simulation

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◦ ◦ Fig. 11. Simulated surface temperature at 85.1 S, 21 W in 2010.

listed in Table I. Moreover, if the slope is shaded by other crater walls of mountains in both winter and summer, T can be less than 40 K as well. In Part 2 ( T > 40 K, ω<0◦),large T on the illuminated slope is likely to be caused by the seasonal shadowing of other mountains or crater walls. Because the incident angle on the polar region is always very large, it might make very large shaded acreage. Even small ﬂuctuation in the incident angle can yield a large change in the shaded area, as shown in No. 2 box of Fig. 8. In Part 3 ( T < 40 K, ω>0◦), the surface with large ω can be shaded in both winter and summer and keeps cold all the year, resulting in small seasonal temperature difference, as shown in No. 3 box of Fig. 8, which locates at the shaded Fig. 12. Diurnal temperature in the PSR. (a) Temperature at daytime crater wall. (dgdr_tb8_avg_poln_20110306d_240_img). The highest temperature in the In Part 4 ( T > 40 K, ω>0◦),theω is positive but small, black box can reach 109 K. (b) Local time of (a). (c) Temperature at night (dgdr_tb8_avg_poln_ 20110320n_240_img). (d) Local time of (c). The useless as shown in the No. 4 box of Fig. 8. The latitude of the place data are shown in white. (e) Temperature difference. The horizontal and ◦ is −82 , making sure the surface will not be shaded by itself vertical axes are pixel number. even in winter. Large T > 40 K might be caused by the seasonal shadow of other crater walls of mountains. ◦ ◦ ◦ the data from March 20, 2011 (subsolar latitude 1.5 N) to In the polar region (85.1 S, 21 W) with the slope angles April 16, 2011 (subsolar latitude 1.4◦ N) at the local time γ =− . ◦ ω =− . ◦ 2 75 and 13 65 , the measured temperature in around 22:30 are used here. March is 255 K at the local time 12:10, and the temperature Fig. 12(a)–(e) shows the temperature data and the local in September is 269 K at the local time 12:15. The selected time. The temperature difference between the daytime and the

region is not shaded both in winter and summer. This T night is shown in Fig. 12(e). It can be found that the largest is 14 K. Fig. 11 shows our simulated temperature of 2010. difference can reach 40 K. The highest temperature in the PSR The simulated temperature is 254 K in March and 269 K of the Hermite-A [in the black box in Fig. 12(a)] can even in September at the corresponding local time. It gives the reach 109 K at 14:30. The edge of the crater wall is exposed = simulated seasonal T 15 K. to illumination at daytime and is of high temperature. The IR radiation of the heated crater walls is the dominant thermal D. Diurnal Temperature Change in PSR source of the PSR [33]. The PSR of the moon poles with no illumination keeps an extremely low temperature all the year. Temperature in some E. Seasonal Temperature Variation in PSR of these PSR can drop as low as 25 K [1]. These regions To see the seasonal change of the temperature in the PSR, are potential deposits for water ice, a 1-mm-thick layer of the Diviner data from May 10, 2010 (the subsolar latitude which can be preserved against sublimation under 100 K for 1.3◦ N) to June 6, 2010 (the subsolar latitude 0.7◦ N) and more than 2 billion years [30]. If there is water ice in PSR, from August 19, 2012 (the subsolar latitude 1.5◦ S) to Sep- e.g., the Hermite-A crater (87.8◦ N 47.1◦ W) has been under tember 15, 2012 (the subsolar latitude 1.4◦ S) at the local time discussion [31], [32]. The temperature of the Hermite-A crater about 09:40 are used. The local time differences in most places can be as low as 35 K [32]. in the crater are within 1 h. The temperature difference T The Diviner IR data of the Hermite-A crater collected from is obtained by subtracting the temperature of 2012 from the March 6, 2011 (subsolar latitude 1.5◦ N) to April 2, 2011 value of 2010. Fig. 13(a)–(d) shows these data, T and the (subsolar latitude 1.5◦ N) at the local time around 14:30 and local time difference.

Fig. 15. Temperature and DEM (ldem_75n_240m_ﬂoat) section of AA at ◦ the North Pole in Fig. 14. The longitude is 53 W.

Fig. 13. Seasonal temperature variation of the Diviner data in the PSR. (a) Diviner data from May 10, 2010 to June 6, 2010 (dgdr_tb8_avg_poln_20100510d _240_img). (b) Diviner data from August 19, 2012 to September15, 2012 (dgdr_tb8_ avg_poln_20120819d_240_img). (c) Temperature difference. (d) Local time difference. The horizontal and vertical axes are pixel number.

Fig. 16. (a) Temperature of the area in the black box in Fig. 14. (b) DEM Fig. 14. Resampled data of Fig. 12(a) into cylindrical projection. of this area. (c) γ of this area. (d) ω of this area.

It can be seen that the largest T is more than 30 K. The in the PSR have higher temperatures. The highest temperature incidence angle of the solar illumination varies from month to along the section AA is 108 K, which locates closer to the month, and affects the temperature of the sunlit crater wall. heat source on the crater wall. It might be the main reason for this T in the PSR. Fig. 16 (a)–(d) shows the temperature, DEM, surface slope angles γ ,andω in the box in Fig. 14. It can be found that the F. Temperature of PSR Versus Topography temperature in the PSR is related to the topography, especially the angle ω. A surface in the PSR facing the sunlit slope is To see the temperature distribution within the PSR, the data with a negative ω. As a result, the temperature increases with in Fig. 12 are resampled into cylindrical projection, as shown the decrease in ω. in Fig. 14. Fig. 15 shows the temperature and altitude of the IV. SEASONAL MICROWAVE TB AT THE POLAR REGION section AA. It can be seen that the temperature peak appears on the sunlit slope, which is the heat source for the PSR. A. MW TB Model On the other side of the crater, the temperature peak is outside The MW radiometers onboard CE-2 acquired MW TB data the rim of the crater. The radiation of this part would be shaded at four frequency channels, i.e., 3, 7.8, 19.3, and 37 GHz. The by the edge of the crater. The slopes facing the heat source measurement time, selenographic longitude, and latitude of the

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data are recorded as well. In this article, TBs of 37 GHz are employed. The lunar regolith is seen as a model of fine-grained soil media over the bedrock. Based on the fluctuation-dissipation theorem and the WKB approach, MW TB from the uniformly inhomogeneous media can be obtained [34], [35]. At nadir observation, it can be simplified as ∞ z − κa(z )dz TB(0) =[1 − R(0)] κa(z)T (z)e 0 dz (15) 0 where R(0)is the reflectivity at nadir observation. There is √ 2 1 − εs R(0) = √ (16) 1 + εs where εs is the dielectric constant of the surface and can be obtained by (18) with ρ(0) of (23). The influence of the surface slopes is trivial and ignorable when the incidence angle is less ◦ than 20 . The absorption coefficient of the regolith media, κa, is derived as 2πvε(z) κa = √ (17) c ε(z) where v is the frequency, c is the light speed in free space, ε(z) and ε(z) are the real and imaginary parts of the dielectric constant of the regolith at the depth z, respectively. ε( )/ε( ) = δ The ratio z z tan is defined as the loss tangent. Fig. 17. (a) Measurement date of the CE-2 data versus local time of the data. The ε (z) can be obtained by the Maxwell–Garnett formula The data between the black dashed lines are picked for study. (b) 37-GHz TB [36] as data, whose local time are between 14:00 and 15:00, are normalized to 14:30. ε− ε( ) − 1 1 = 1 z 1 TABLE II (18) ρ ε+2 ρ(z) ε (z) + 2 SEASONAL TB VARIATION FROM CE-2 DATA where the normalized ε=2.75 with ρ=1.7g/cm3.The dielectric constant ε(z) of the regolith media is determined by (18) when ρ(z) is obtained by (23). The loss tangent is fitted by the CE-2 TB37 data against TiO2 abundance as [15] −4 tan δ = 3.516 × 10 TiO2 + 0.0087 as TiO2 > 1% (19a) −5 tan δ =−8.945 × 10 TiO2 + 0.0097 as TiO2 < 1%. (19b) B. Analysis of MV TB37 Data ◦ ◦ The density profile was fitted by the measurements of CE-2 TB37 data within the latitude range of (75 S–74.5 S) the regolith samples collected at the Apollo landing sites are used here. Fig. 17(a) and (b) shows the local time, date of (specifically indicated by the subscript 0) as [36] observation, and TB37 data from October to November 2010 (subsolar latitude 1.5◦ S–0.7◦ S) and from April to May 2011 z + 0.122 ◦ ◦ ρ (z) = . . (subsolar latitude 1.5 N–0.4 N). The local time is con- 0 1 919 + . (20) z 0 18 strained from 14:00 to 15:00. Following [39], these data The porosity profile of the regolith is usually written as [2] are all normalized to 14:30. The normalized TB data along

n(z) = 1 − ρ0(z)/G0 (21) the longitude are shown in Fig. 17(b). It can be seen that TB of October and November are higher than the TB of = . 3 37 37 where G0 3 1g/cm is the nominal specific gravity (bulk April and May. Most of the TB differences are in the range of density without void). 5–10 K. The specific gravity is fitted against the mineral abundance Table II shows the TB37 data of two pixels in Fig. 17. The of lunar samples [37] track numbers of the data are 0436 and 2601. The distance G = 27.3FeO + 11TiO2 + 2773 (22) between the centers of two observations is about 3 km, and the local time difference is 0.13 h. The distance and the local where G is in kg/m3. FeO and TiO are the percentage of 2 time difference between two selected TBs in Table II are very mineral by weight, which can be derived from Clementine small to minimize the influence of topography and local time. data [38]. The density profile ρ(z) at different places across The seasonal TB difference is 7 K. the lunar surface can be written as According to the DEM data, the average slope in this area ρ(z) = (1 − n)G. (23) is ω =−0.07◦ and γ = 0.2◦. Based on the algorithm in [38]

◦ Fig. 19. Inﬂuence of the slopes on the 37-GHz TB in 2010 at 74.4 S, ◦ 135 E.

TABLE IV SEASONAL TB37 VARIATION IN THE PSR

◦ ◦ Fig. 18. (a) Simulated 37-GHz TB at 74.4 S, 135 E in 2010 and 2011. ◦ (b) Simulated 37-GHz TB at November 2010 and May 2011 at 74.4 S, ◦ 135 E. crater (87◦ S, 5.1◦ W). Since the location in both latitude and TABLE III longitude of these two observation centers is very close, with ◦ ◦ a distance of 1 km, the effect of the topographic difference SIMULATED TB37 AT 74.4 S, 135 E between these two observations is ignored. The local time difference of these two observations is about 0.1 h. The ◦ ◦ subsolar latitude is about 0.8 S and 1.4 N. The TB37 difference is 15.8 K. The seasonal effect on TB37 can be found in other places in PSR as well. When ﬁtting the TB37 at the polar region, it is important to consider the seasonal effect.

V. C ONCLUSION = . and the Clementine UVVIS data, there is FeO 6 6% and A small tilt in the spin axis of the moon causes different = . TiO2 0 4%. incidence angles of solar illumination and seasonal tempera- Fig. 18(a) shows the simulated TB37 of 2010 and 2011 in ture variation on the lunar polar region. Based on the position

this region. The seasonal difference TB between highest of the subsolar point, the real-time model is presented to cal- temperatures is about 10 K, while the difference between the culate the incidence angle of illumination all the year. Solving lowest temperatures is about 6 K. the 1-D heat conduction equation, the seasonal temperature of Fig. 18(b) shows that the simulated TB37 in Novem- the lunar surface is simulated and validated with the Diviner ber 2010 is 176 K and in May 2011 is 166 K at the IR data. It is seen that the temperature in the PSR is seasonal corresponding local time. The simulated result is close to the as well. With the ﬂuctuation-dissipation theorem and the WKB measured date in Table II. approach, the seasonal MW TB of the lunar regolith is also Fig. 19 shows the simulated TB37 for different slope angles simulated and validated by the CE-2 TB data. The conclusions ω ◦ ω . It can be found that a decrease of 5 in may lead to an are as follows. increase of more than 10 K in TB . The seasonal differences 37 1) The seasonal temperature variation on the lunar polar TB range from 6.5 to 10.7 K for different values of ω, 37 region is related to the lunar topography, i.e., sur- as shown in Table III. When the surface is shaded slopes by face slopes. A high temperature of the sunlit slopes other crater walls seasonally, the temperature difference would corresponds to a small seasonal temperature variation. be much large. A small slope angle corresponds to evident temperature variation. C. Analysis of MV TB37 Data in PSR 2) There are even seasonal temperature variation and diur- It is found that the MW TB37 in the PSR has seasonal nal variation in the PSR. This is affected by the tem- variation as well. As example, Table IV shows the selected perature variation of the surrounding crater walls under CE-2 TB37 data of the tracks 0598 and 1683 in the Haworth illumination.

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3) The MW TB also shows seasonal TB variation at the [16] J. Meeus, Astronomical Algorithms. Richmond, VA, USA: Willmann- lunar polar region, both in and out the PSR. Bell, 1991, pp. 314–347. [17] F. Yang, Y. Xu, K. L. Chan, X. Zhang, G. Hu, and Y. Li, “Study of 4) It is found that the highest temperature data in the PSR Chang’E-2 microwave radiometer data in the lunar polar region,” Adv. of Hermite-A crater collected at March 2011 can reach Astron., vol. 2019, Apr. 2019, Art. no. 3940837. 109 K, which is very close to 110 K. [18] T. A. Mutch, Geology of the Moon: A Stratigraphic View. Princeton, NJ, USA: Princeton Univ., 1972, p. 26. The influence of the seasonal and diurnal temperature [19] G. D. Racca, “Moon surface thermal characteristics for moon orbit- variation on the retrieval of the potential deposits for water ice ing spacecraft thermal analysis,” Planet. Space Sci., vol. 43, no. 6, requires improved modeling and more truth-measurements. pp. 835–842, 1995. [20] R. B. Roger, D. M. Donald, and E. W. Jerry, Fundamentals of Astrody- namics. New York, NY, USA: Dover, 1971. ACKNOWLEDGMENT [21] M. Chapront and J. Chapront, “The lunar ephemeris ELP 2000,” Astron. The authors would like to thank the anonymous reviewers Astrophys., vol. 124, pp. 50–62, Jul. 1983. [22] S. Yu and W. Fa, “Thermal conductivity of surficial lunar regolith for their helpful comments and the Chang’e, LRO, and Diviner estimated from Lunar Reconnaissance Orbiter Diviner Radiometer data,” operations teams for the collection of the high-quality data sets Planet. Space Sci., vol. 124, pp. 48–61, May 2016. used in this article. The data used in this article are publicly [23] M. J. Ledlow, J. O. Burns, G. R. Gisler, J.-H. Zhao, M. Zeilik, and D. N. Baker, “Subsurface emissions from Mercury-VLA radio obser- available via Information system of China’s Lunar Explo- vations at 2 and 6 centimeters,” Astrophys. J., vol. 384, pp. 640–655, ration Program (http://moon.bao.ac.cn/) and the Geosciences Jan. 1992. Node of the Planetary Data System (LOLA: http://pds- [24] B. S. Hemingway, K. M. Krupka, and R. A. Robie, “Heat capacities of the alkali feldspars between 350 and 1000 K from differential scanning geosciences.wustl.edu/missions/lro/lola.htm; Diviner: http:// calorimetry, the thermodynamic functions of the alkali feldspars from pds-geosciences.wustl.edu/missions/lro /diviner.htm). 298.15 to 1400 K, and the reaction quartz + jadeite = analbite,” Amer. Mineralogist, vol. 66, nos. 11–12, pp. 1202–1215, 1981. REFERENCES [25] J. A. Fountain and E. A. West, “Thermal conductivity of particulate basalt as a function of density in simulated lunar and Martian envi- [1] K.-M. Aye, D. A. Paige, M. C. Foote, B. T. Greenhagen, ronments,” J. Geophys. Res., Planets, vol. 75, no. 20, pp. 4063–4069, and M. A. Siegler, “The coldest place on the moon,” in 1970. Proc. Lunar Planet. Sci. Conf., 2013. [Online]. Available: [26] J. L. Bandfield, P. O. Hayne, J.-P. Williams, B. T. Greenhagen, and https://www.lpi.usra.edu/meetings/lpsc2013/eposter/3016.pdf D. A. Paige, “Lunar surface roughness derived from LRO Diviner [2]G.H.Heiken,Lunar Sourcebook: A User’s Guide to the Radiometer observations,” Icarus, vol. 248, pp. 357–372, Mar. 2015. Moon. New York, NY, USA: Cambridge Univ. Press, 1991, [27] S. J. Keihm, “Interpretation of the lunar microwave brightness tem- pp. 285–356. perature spectrum: Feasibility of orbital heat flow mapping,” ICARUS, [3] M. Siegler, D. Paige, J.-P. Williams, and B. Bills, “Evolution of lunar vol. 60, no. 3, pp. 568–589, 1984. polar ice stability,” Icarus, vol. 255, pp. 78–87, Jul. 2015. [28] J.-P. Williams, D. A. Paige, B. T. Greenhagen, and E. Sefton-Nash, [4] H. Noda et al., “Illumination conditions at the lunar polar regions “The global surface temperatures of the moon as measured by the by KAGUYA(SELENE) laser altimeter,” Geophys. Res. Lett., vol. 35, diviner lunar radiometer experiment,” Icarus, vol. 283, pp. 300–325, no. 24, Dec. 2008, Art. no. L24203, doi: 10.1029/2008GL035692. Feb. 2017. [5]D.B.J.Busseyet al., “Illumination conditions of the south pole of [29] D. A. Paige et al., “LRO diviner lunar radiometer global mapping results the Moon derived using Kaguya topography,” Icarus, vol. 208, no. 2, and gridded data product,” in Proc. 42nd Lunar Planet. Sci. Conf., pp. 558–564, Aug. 2010. Mar. 2011, p. 2544. [6] E. Mazarico, G. A. Neumann, D. E. Smith, M. T. Zuber, and [30] E. A. Fisher et al., “Evidence for surface water ice in the lunar M. H. Torrence, “Illumination conditions of the lunar polar regions using polar regions using reflectance measurements from the lunar orbiter LOLA topography,” Icarus, vol. 211, no. 2, pp. 1066–1081, Feb. 2011. Laser Altimeter and temperature measurements from the diviner lunar [7] D. A. Paige et al., “The lunar reconnaissance orbiter diviner lunar radiometer experiment,” Icarus, vol. 292, pp. 74–85, Aug. 2017. radiometer experiment,” Space Sci. Rev., vol. 150, pp. 125–160, [31] N. Liu, W. Fa, and Y.-Q. Jin, “No water–ice invertable in PSR Jan. 2010. of Hermite-a crater based on mini-RF data and two-layers model,” [8]P.O.Hayneet al., “Global regolith thermophysical properties of the IEEE Geosci. Remote Sens. Lett., vol. 15, no. 10, pp. 1485–1489, Moon from the Diviner Lunar Radiometer Experiment,” J. Geophys. Oct. 2018. Res., Planets, vol. 122, no. 12, pp. 2371–2400, 2017. [32] O. P. N. Calla, S. Mathur, and K. L. Gadri, “Quantification of water ice [9] A. R. Vasavada, “Lunar equatorial surface temperatures and regolith in the Hermite-A crater of the lunar north pole,” IEEE Geosci. Remote properties from the diviner lunar radiometer experiment,” J. Geo- Sens. Lett., vol. 13, no. 7, pp. 926–930, Jul. 2016. phys. Res., Planets, vol. 117, no. E12, Dec. 2012, Art. no. E00H18, [33] D. A. Paige, “Diviner lunar radiometer observations of cold traps doi: 10.1029/2011JE003987. in the Moon’s south polar region,” Science, vol. 330, no. 6003, [10] J. P. Williams et al., “Seasonal variations in south polar tempertures on pp. 479–482, Oct. 2010. the moon,” in Proc. 50th Lunar Planet. Sci. Conf., 2019, pp. 1–2. [34] Y.-Q. Jin, Electromagnetic Scattering Modelling for Quantitative Remote [11] W. Fa and Y.-Q. Jin, “A primary analysis of microwave brightness Sensing. Singapore: World Scientific, 1993, p. 310. temperature of lunar surface from Chang-E 1 multi-channel radiometer [35] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote observation and inversion of regolith layer thickness,” Icarus, vol. 207, Sensing Fundamentals and Radiometry. Microwave Remote Sensing no. 2, pp. 605–615, Jun. 2010. Active & Passive, vol. 1. New York, NY, USA: Addison-Wesley, 1981, [12] Y.-Q. Jin and W. Fa, “The modeling analysis of microwave emission pp. 229–230. from stratified media of nonuniform lunar cratered terrain surface for [36] W. Fa and M. A. Wieczorek, “Regolith thickness over the lunar nearside: chinese chang-E 1 observation,” IEEE Geosci. Remote Sens. Lett.,vol.7, Results from Earth-based 70-cm Arecibo radar observation,” Icarus, no. 3, pp. 530–534, Jul. 2010. vol. 218, no. 2, pp. 771–787, 2012. [13] T. Fang and W. Fa, “High frequency thermal emission from the lunar [37] Q. Huang and M. A. Wieczorek, “Density and porosity of the lunar surface and near surface temperature of the Moon from Chang’E-2 crust from gravity and topography,” J. Geophys. Res., Planets, vol. 117, microwave radiometer,” Icarus, vol. 232, pp. 34–53, Apr. 2014. no. E5, May 2012, Art. no. E05003, doi: 10.1029/2012JE004062. [14] N. Liu and Y.-Q. Jin, “A radiative transfer model for MW cold and IR [38] P. G. Lucey, “Lunar iron and titanium abundance algorithms based on hot spots of Chang’e and diviner observations,” IEEE Trans. Geosci. final processing of Clementine ultraviolet-visible images,” J. Geophys. Remote Sens., vol. 57, no. 10, pp. 8184–8190, Oct. 2019. Res., Planets, vol. 105, no. E8, pp. 20297–20305, Aug. 2000. [15] N. Liu, W. Fa, and Y.-Q. Jin, “Brightness temperature of lunar surface [39] Y.-C. Zheng, K. T. Tsang, K. L. Chan, Y. L. Zou, F. Zhang, and for calibration of multichannel millimeter-wave radiometer of geosyn- Z. Y. Ouyang, “First microwave map of the moon with Chang’E-1 chronous FY-4M,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 5, data: The role of local time in global imaging,” Icarus, vol. 219, no. 1, pp. 3055–3063, May 2019. pp. 194–210, May 2012.

Niutao Liu was born in Jiangsu, China, in 1994. of Electromagnetic Waves (MoE), and Institute of EM Big Data and He received the B.S. degree from the Nangjing Uni- Intelligence Remote Sensing, Fudan University, Shanghai, China. He has versity of Posts and Telecommunications, Nanjing, authored more than 780 articles in refereed journals and conference proceed- China, in 2016. He is currently pursuing the Ph.D. ings and 14 books, including Polarimetric Scattering and SAR Information degree from the School of Information Science and Retrieval (Wiley and IEEE, 2013), Theory and Approach of Information Engineering, Fudan University, Shanghai, China. Retrievals From Electromagnetic Scattering and Remote Sensing (Springer, His research interests include planetary remote 2005), and Electromagnetic Scattering Modelling for Quantitative Remote sensing, computational electromagnetics, and target Sensing (World Scientiﬁc, 1994). His research interests include electromag- recognition. netic scattering and radiative transfer in complex natural media, microwave satellite-borne remote sensing, and theoretical modeling, information retrieval and applications in earth terrain and planetary surfaces, and computational electromagnetics. Dr. Jin was a member of the IEEE GRSS AdCom and the IEEE GRSS Major Awards Committee. He is a fellow of the World Academy of Sciences Ya-Qiu Jin (SM’89–F’04–LF’18) received the B.S. for Advances of Developing World (TWAS) and the International Academy of degree in atmospheric physics from Peking Univer- Astronautics (IAA). He received the IEEE GRSS Distinguished Achievement sity, Beijing, China, in 1970, and the M.S., E.E., Award in 2015, the IEEE GRSS Education Award in 2010, the China National and Ph.D. degrees in electrical engineering and Science Prize in 2011 and 1993, the Shanghai Sci/Tech Gong-Cheng Award computer science from the Massachusetts Institute of in 2015, the ﬁrst-grade MoE Science Prizes in 1992, 1996, and 2009 among Technology, Cambridge, MA, USA in 1982, 1983, many other prizes. He was awarded the Senior Research Associateship in and 1985, respectively. NOAA/NESDIS by the U.S. National Research Council in 1996. He is He was a Research Scientist of Atmospheric an Academician of Chinese Academy of Sciences (CAS). He is an IEEE Environmental Research (AER), Inc., Cambridge, GRSS Distinguished Speaker and an Associate Editor of IEEE ACCESS. in 1985, the Research Associate Fellow with the City He was the Co-Chair of TPC for IGARSS2011 in Vancouver, BC, Canada, University of New York, New York, NY, USA, from and the Co-General Chair for IGARSS2016 in Beijing, China. He was 1986 to 1987, and a Visiting Professor with the University of York, York, an Associate Editor of the IEEE TRANSACTIONS ON GEOSCIENCE AND U.K., in 1993, sponsored by the U.K. Royal Society. He is currently a Te-Pin REMOTE SENSING from 2005 to 2012, and the Chair of the IEEE Fellow Professor and the Director of the Key Laboratory for Information Science Evaluation of GRSS from 2009 to 2011.

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