Spatio-Temporal Characteristics for Moon-Based Earth Observations

: Spatio-temporal characteristics are the crucial conditions for Moon-based Earth observations. In this study, we established a Moon-based Earth observation geometric model by considering the intervisibility condition between a Moon-based platform and observed points on the Earth, which can analyze the spatio-temporal characteristics of the observations of Earth’s hemisphere. Furthermore, a formula for the spherical cap of the Earth visibility region on the Moon is analytically derived. The results show that: (1) the observed Earth spherical cap has a diurnal period and varies with the nadir point. (2) All the annual global observation durations in di ﬀ erent years show two lines that almost coincide with the Arctic circle and the Antarctic circle. Regions between the two lines remain stable, but the observation duration of the South pole and North pole changes every 18.6 years. (3) With the increase of the line-of-sight minimum observation elevation angle, the area of an intervisible spherical cap on the lunar surface is obviously decreased, and this cap also varies with the distance between the barycenter of the Earth and the barycenter of the Moon. In general, this study reveals the e ﬀ ects of the elevation angle on the spatio-temporal characteristics and additionally determines the change of area where the Earth’s hemisphere can be observed on the lunar surface; this information can provide support for the accurate calculation of Moon-based Earth hemisphere observation times.


Introduction
In recent years, Earth system science has considered the planet as an indivisible whole, with large-scale geoscience phenomena such as global warming receiving increasing attention. To understand, monitor, and predict global change, many countries and organizations have created three global observation systems. These systems are named the Global Climate Observing System (GCOS), the Global Ocean Observing System (GOOS), and the Global Terrestrial Observing System (GTOS), and their objectives are to observe the climate, oceans, and land [1][2][3]. Taking the GCOS as an example, it is designed to detect climate change, monitor the climate system, and perform climate forecasting at a global scale [4]. The European Space Agency (ESA) also proposed a project called Long-Term Data Preservation (LTDP), whose mission includes recording regional to global Earth Earth observations. Song et al. [19] indicate that the lunar polar regions are the best regions to build a Moon-based observatory because the variation in lunar night temperature is no more than 0.2 K.
To prove that a Moon-based platform has the ability to observe a large-scale phenomenon, from the perspective of viewing geometry, Ren et al. [20] proposes a new concept of effective coverage, and demonstrates that for almost 75% of the area of the near Earth side on the lunar surface, the average effective coverage rate approaches nearly one-half. Ye et al. [21] proposes a geometry model based on the model of Ren et al. then calculates parameters that are related to the observation scope. Their results enhance our understanding of the potential for observing large-scale geoscientific phenomena. Regarding the observation duration, Ye et al. [22] reveals that the global annual observation duration from selenocenter is approximately 90 days, increasing from north to south in 2016. Wherever a Moon-based platform would be, when observing 0 • latitude Earth surface features, observation durations would all be more than 40 days during one year.
Conferences have also been held in recent years to demonstrate the progress of research on Moon-based observations. The First Symposium on Space Earth Science (SESS 2018) was held in Sanya, China, and one of its sessions was focused on Moon-based Earth observations; some experts, from the perspective of the preliminary design of the payload, detailed outgoing radiation from a Moon-based platform for Earth observations, the design of a Moon-based SAR Earth observation system, the effects of lunar terrain on Moon-based observations, and so on in order to share the progress of their studies. In 2019, the First China Space Science Assembly (CSSA) was held with the theme of "Develop Space Science and Build Aerospace Power." As a potential means for Earth observations, Moon-based Earth observations has attracted considerable attention. Scholars in the aspects of Moon-based infrared surface temperature inversion, a data transmission scheme design of a Moon-based platform, and global cloud distribution extraction from a simulation image, introduced their research.
The aim of this study is to define the spatio-temporal characteristics of a Moon-based platform. We make two contributions in this paper. First, according to the unique observation geometry of a Moon-based platform, we find the period of the observed Earth spherical cap. When compared with the annual global observation duration of different years, the observation duration distribution varies with the latitude of the nadir point. Second, we derive the analytical formula of the intervisible spherical cap of the Moon as suitable places to equip sensors that can observe the Earth's entire hemisphere, and calculate its area. The area is in connection with the line-of-sight minimum observation elevation angle and distance between the barycenter of the Earth and the barycenter of Moon (the barycenter separation distance); thus, it can support calculating the accurate time for Moon-based Earth hemisphere observations.

Coordinate System Definitions and Moon-Based Earth Observation Geometry
In this section, we briefly introduce the astronomical coordinate systems used in this study. Because the Moon and the Earth are in motion in their respective orbits, the positions of the features on the Earth are defined in the geodetic coordinate system of the Earth and the positions of a Moon-based sensor are defined in the geodetic coordinate system of the Moon. Thus, to elucidate the spatio-temporal characteristics for Moon-based Earth observations, we should transform the Earth and the Moon into a unified coordinate system. The five main coordinate systems used are as follows: 1.
International Terrestrial Reference System (ITRS). This is the fundamental coordinate system we have chosen to unify the Earth and the Moon. Specifically, the x-axis of ITRS points to (0 • , 0 • ) in the latitude and longitude network coordinate system, the z-axis is oriented toward the north pole, and the y-axis completes the rotation in the right-handed Cartesian system [23].

2.
Geocentric Celestial Reference System (GCRS) and International Celestial Reference System (ICRS). What makes these systems unique are their origins, as GCRS is a geocentric coordinate To calculate the observation scope, the first step is to determine the geometric relationship of the Earth and the Moon.
The geometric relationship is shown in Figure 1. The barycenter of the Earth and the Moon are presented as O E and O M , respectively. Surface features on the Earth are shown as F, point P is the position of a Moon-based platform, and S is the position of the Sun. The Earth's and a Moon-based platform position vectors are designated R F and R P , respectively, and the barycenter vector of the Earth and the Moon are set to R E and R M , respectively; v is the vector from a Moon-based platform to the Earth surface features, which can be expressed as follows R F and R P are calculated by where vector e ITRS represents the Earth surface features' position vector in ITRS, and m MCMF is the position vector of a  [28,29]. Next, we detail how to unify the Earth and the Moon in the same coordinate system. The initial coordinates of the Earth's barycenter are obtained from development ephemeris (DE), which is studied by the Jet Propulsion Laboratory (JPL) regarding the ICRS. JPL DE has been widely used in astrogation, planetary exploration and analysis, and reductions of most astronomical precision observations, and has provided data for our study [30]. There have been many versions of the JPL DE, with each ephemeris produced by numerical integration of the equations-i.e., Chebyshev polynomial of motion. In this study, we use DE 430, which adds lunar laser ranging data and is the most accurate lunar ephemeris that includes both libration and nutation [31]. parameters (EOP) that include information of the Earth's rotation. [P] and [N] are called the precession matrix and nutation matrix, respectively, and reveal the precession and nutation characteristics of the Earth, respectively. [R] is the rotation matrix and represents the irregularities of the rotation angle.
[M] and [B] are the polar motion matrix and bias matrix, respectively. Matrix [L] represents the lunar libration with three Euler angles that are also derived from planetary ephemeris, and matrix [C] is the constant matrix used to transform ME to PA, because data from planetary ephemeris are defined in the ME and their rotation axes are not the same [28,29]. The transformation of the Earth is from ICRS to the ITRS. In the inertial reference system, ICRS to GCRS can be understood as the transformation from the solar system's barycenter to the Earth's barycenter. GCRS to ITRS is a more complicated transition, requiring EOP as mentioned above [32]. Additionally, a Moon-based platform can be indicated in the SCS. Transforming it into the MCMF means that the platform can be expressed as a Cartesian coordinate system. To describe its orientation, the transformation from MCMF to ICRS requires Euler angles of the lunar libration, which are derived from JPL DE 430. Then, the remaining transformations of the Moon from ICRS to ITRS are the same as the Earth.
Assuming (λ F , ϕ F ) as the coordinates of a feature on the Earth's surface, then the coordinates of this feature in ITRS can be written as: where r E is the radius of the Earth. Similarly, assuming (λ L , ϕ L ) as the coordinates of a Moon-based platform in the SCS, the coordinates transformed from the SCS to ME system are calculated as: where r M is the radius of the Moon and (x M , y M , z M ) are the coordinates of a Moon-based platform in the ME coordinate system. For the transformation of a Moon-based platform from the ME reference system to ITRS, the transformation process is expressed as [14]: where P is a Moon-based platform's position, matrix [T] denotes a translation matrix; regardless of which vector is multiplied by [T], the vector will not change its length and orientation but be translated. The coordinates of the Moon's barycenter are the elements in the translation matrix that can be obtained from planetary ephemeris. The remaining seven matrixes were introduced at the beginning of this section. Figure 2 shows the Earth spherical cap viewed from a Moon-based sensor. The tangent line starts at a Moon-based platform that intersects the spherical surface of the Earth at A E and B E . Considering the minimum observation elevation angle of the Earth ε min (it is the critical angle which allows the line-of-sight from a Moon-based sensor that can reach to the observed features on the Earth, breaking Remote Sens. 2020, 12, 2848 6 of 18 through the limit of Earth's local terrain, shown in Figure 2), then the Earth spherical cap changes from A E N E B E to A E N E B E , which can be expressed as: where h is the height of Earth spherical cap, representing the distance between point G E and N E in Figure 2. sin(π/2+ε min ) where d is the distance between a Moon-based platform and the Moon's nadir point on the Earth (Earth-Moon distance). α is the angle between line MA E and line MO E . Line MA E starts from the position of a Moon-based platform (point M) and ends at the tangent line considering the minimum observation elevation angle ε min , shown as A E in Figure 2. Line MO E is the line that connects point M and the barycenter of the Earth O E . Thus, because the angle α is far less than π/2, Equation (6) can be simplified as:  The Earth spherical cap is connected with Earth-Moon distance d and minimum observation elevation angle .  The Earth spherical cap is connected with Earth-Moon distance d and minimum observation elevation angle ε min .

Determination of Intervisibility Conditions
where d E (d M ) represents the distance between the barycenter of the Earth O E (Moon O M ) and point C.
In addition: D is the barycenter separation distance. By substituting (10) into (9), thus leading to: If we consider that the line-of-sight minimum observation elevation angle θ min on the lunar surface, which partly depends on the lunar terrain, is the prerequisite for the line-of-sight vector breaks through the limit of lunar local terrain, the lunar intervisible spherical cap is changed to A M N M B M , and ψ M (angle between line A M O M and line O M C) can be calculated by: Simplifying Equation (12), we can derive: Thus, Remote Sens. 2020, 12, x FOR PEER REVIEW 7 of 18 where dE (dM) represents the distance between the barycenter of the Earth OE (Moon OM) and point C.
In addition: D is the barycenter separation distance. By substituting (10) into (9), thus leading to: If we consider that the line-of-sight minimum observation elevation angle θmin on the lunar surface, which partly depends on the lunar terrain, is the prerequisite for the line-of-sight vector breaks through the limit of lunar local terrain, the lunar intervisible spherical cap is changed to Simplifying Equation (12), we can derive: Thus, where (x, y, z) represent coordinates of a Moon-based platform. The nadir point NM meets the demand of the equation when ignoring the ellipticity of the Earth, as shown by: By substituting (16) into (15), we can derive that:  where (x, y, z) represent coordinates of a Moon-based platform. The nadir point N M meets the demand of the equation when ignoring the ellipticity of the Earth, as shown by: By substituting (16) into (15), we can derive that: By replacing (14) to (17), the final criterion can be written as: Thus, the hemisphere of the Earth can be seen by a Moon-based sensor when Equation (18) where H is the height of the spherical cap, representing the distance between points G M and N M in Figure 3. Through the derivation, we find that the lunar intervisible spherical cap area is related to the line-of-sight minimum observation elevation angle θ min and the barycenter separation distance D.
When the barycenter separation distance D continuously changes, the variation rate of the lunar intervisible spherical cap can be written as: Because of: Therefore, Similarly, in considering the derivative of the line-of-sight minimum observation elevation angle θ min with respect to the lunar intervisible spherical cap, thus leading to: Because the sum of radii of the Earth and the Moon is much smaller than the barycenter separation distance, as the distance increases, the rate of lunar intervisible spherical cap size change becomes smaller.

Spatio-Temporal Coverage Performance
To elucidate the global observation coverage of the Earth, we first assume a Moon-based sensor is located at the (0 • , 0 • ) on the lunar surface to observe the Earth. Figure 4 is the global observation coverage at corresponding moments of the first two days in 2021. Thus, the observation scope is connected with the angle of the orbit of the Moon to the ecliptic and the obliquity of the ecliptic. Furthermore, the distributions of the observed area at the same moment from two different days appear alike. Thus, the Earth observation shows daily periodicity. To determine the regularity of the observation coverage, we calculate the variation of the Earth spherical cap with d and ε min , the three-dimensional curve line is shown in Figure 5. We choose the variation of d E in January 2021, and ε min is continuously changing from 0 • to 40 • . The largest Earth spherical cap can reach 2.51 × 10 8 km 2 and the smallest Earth spherical cap is approximately 8.8 × 10 7 km 2 . Obviously, a 27-day cycle occurs that corresponds to the lunar orbit period. Notably, the valley and peak of the curve correspond to the perigee and the apogee of the lunar orbit. Thus, the Earth spherical cap is related to the Earth-Moon distance and the minimum observation elevation angle.
Remote Sens. 2020, 12, x FOR PEER REVIEW 9 of 18 To elucidate the global observation coverage of the Earth, we first assume a Moon-based sensor is located at the (0°, 0°) on the lunar surface to observe the Earth. Figure 4 is the global observation coverage at corresponding moments of the first two days in 2021. Thus, the observation scope is connected with the angle of the orbit of the Moon to the ecliptic and the obliquity of the ecliptic. Furthermore, the distributions of the observed area at the same moment from two different days appear alike. Thus, the Earth observation shows daily periodicity. To determine the regularity of the observation coverage, we calculate the variation of the Earth spherical cap with d and , the threedimensional curve line is shown in Figure 5. We choose the variation of dE in January 2021, and is continuously changing from 0° to 40°. The largest Earth spherical cap can reach 2.51 × 10 8 km 2 and the smallest Earth spherical cap is approximately 8.8 × 10 7 km 2 . Obviously, a 27-day cycle occurs that corresponds to the lunar orbit period. Notably, the valley and peak of the curve correspond to the perigee and the apogee of the lunar orbit. Thus, the Earth spherical cap is related to the Earth-Moon distance and the minimum observation elevation angle.   In the most ideal case, the minimum elevation angle can be equal to 0 • . However, because it is affected by the terrain and surface features which cannot be ignored, the minimum elevation angle is non-zero values. We set this angle to be 10 • , 20 • , and 30 • . Figure 6a shows with the increase of the Earth-Moon distance, the area of the Earth spherical cap is also increased because it is inversely proportional to the Earth-Moon distance. Additionally, the slope of the curve line is tiny and becomes smaller with the increase of the Earth-Moon distance. Figure 6b indicates that with an increase of the minimum observation elevation angles, the Earth spherical cap would decrease. Moreover, if the minimum observation elevation angle increases by the same size, the decrease of the Earth spherical cap would become less intense. The three lines show that the increase of the Earth spherical cap is more sensitive when the elevation angle is small. In general, the Earth spherical cap is relative to the Earth-Moon distance and the minimum observation elevation angle, and with the increase of the distance, the Earth spherical cap also increases. Conversely, with an increase of the minimum elevation angle, the cap decreases. However, the effects of the minimum elevation angle to the Earth spherical cap are much larger than the Earth-Moon distance, which cannot be ignored. The similarity is that with increases of these two factors, both rate changes of the Earth spherical cap will become slower.   Figure 6 displays influences of Earth-Moon distance and minimum observation elevation angle on the Earth spherical cap. In the most ideal case, the minimum elevation angle can be equal to 0°. However, because it is affected by the terrain and surface features which cannot be ignored, the minimum elevation angle is non-zero values. We set this angle to be 10°, 20°, and 30°. Figure 6a shows with the increase of the Earth-Moon distance, the area of the Earth spherical cap is also increased because it is inversely proportional to the Earth-Moon distance. Additionally, the slope of the curve line is tiny and becomes smaller with the increase of the Earth-Moon distance. Figure 6b indicates that with an increase of the minimum observation elevation angles, the Earth spherical cap would decrease. Moreover, if the minimum observation elevation angle increases by the same size, the decrease of the Earth spherical cap would become less intense. The three lines show that the increase of the Earth spherical cap is more sensitive when the elevation angle is small. In general, the Earth spherical cap is relative to the Earth-Moon distance and the minimum observation elevation angle, and with the increase of the distance, the Earth spherical cap also increases. Conversely, with an increase of the minimum elevation angle, the cap decreases. However, the effects of the minimum elevation angle to the Earth spherical cap are much larger than the Earth-Moon distance, which cannot be ignored. The similarity is that with increases of these two factors, both rate changes of the Earth spherical cap will become slower. Next, we quantify the normalized global observation duration at different years to demonstrate the Moon-based Earth observation characteristics, which is shown in Figure 7. From the six subfigures in Figure 7, the observation duration distributes with latitude. We also observe that there are two distinct lines whose range is nearly from 61° to 71°, which nearly match the Arctic circle and Antarctic circle. This is complementary to the angle between the Earth equatorial plane and the ecliptic plane, whose range is from 18.5° to 28.5°. Regions from the top line to 90°N are in the range of the north pole; similarly, regions between the bottom line to 90°S are in the scope of the south pole. The widths of two regions clearly change together when the difference in duration distribution becomes larger; for example, in Figure 7e, the widths of the two regions are smaller than those in Figure 7b, whose duration distribution is relatively average. Moreover, the widths of the two regions change with the maximum latitude of the nadir point in a year. For example, the widths of the two polar regions reach the minimum in 2035 when the maximum latitude of nadir point is smallest in the six years of the subfigures. The regions between two lines remain stable during the observation Next, we quantify the normalized global observation duration at different years to demonstrate the Moon-based Earth observation characteristics, which is shown in Figure 7. From the six subfigures in Figure 7, the observation duration distributes with latitude. We also observe that there are two distinct lines whose range is nearly from 61 • to 71 • , which nearly match the Arctic circle and Antarctic circle. This is complementary to the angle between the Earth equatorial plane and the ecliptic plane, whose range is from 18.5 • to 28.5 • . Regions from the top line to 90 • N are in the range of the north pole; similarly, regions between the bottom line to 90 • S are in the scope of the south pole. The widths of two regions clearly change together when the difference in duration distribution becomes larger; for example, in Figure 7e, the widths of the two regions are smaller than those in Figure 7b, whose duration distribution is relatively average. Moreover, the widths of the two regions change with the maximum latitude of the nadir point in a year. For example, the widths of the two polar regions reach the minimum in 2035 when the maximum latitude of nadir point is smallest in the six years of the subfigures. The regions between two lines remain stable during the observation period. With an increase in observation duration in one polar region, the observation duration of another polar region decreases. Specifically, the observation durations of two polar regions are inversely proportional. This inversion is mainly a product of the geometric relationship between the Earth's surface features, the Sun, and the Moon-based platform, and it is also in connection with the latitude of the nadir point. Moreover, the distribution of the observation duration appears opposite when the time interval is four years, as shown in Figure 7a,b,e,f. However, if the time interval is two years, the distribution is stable, as shown in Figure 7c,d. This is because the 18.6-year lunar nodal precession associated with the Earth-Moon system causes a systematic variation of the Moon's orbit around the Earth. Earth's surface features, the Sun, and the Moon-based platform, and it is also in connection with the latitude of the nadir point. Moreover, the distribution of the observation duration appears opposite when the time interval is four years, as shown in Figure 7a,b,e,f. However, if the time interval is two years, the distribution is stable, as shown in Figure 7c,d. This is because the 18.6-year lunar nodal precession associated with the Earth-Moon system causes a systematic variation of the Moon's orbit around the Earth.

Spatio-Temporal Coverage Performance at Different Positions on the Lunar Surface
In this section, we first introduce the effects of barycenter separation distance and the line-ofsight minimum observation elevation angle of the lunar surface on the lunar intervisible spherical cap. Next, we assume that a Moon-based platform is equipped at different positions on the lunar surface to determine the observation performance from different positions on the Moon. Figure 8 displays the influences of barycenter separation distance and the line-of-sight minimum observation elevation angle on the lunar intervisible spherical cap. As described in [33], in the full observation regions (longitude between 80° W-80°E and latitude between 80° S-80°N), which is less affected by lunar terrain, the line-of-sight minimum elevation angle should be larger than 20°. Thus, we choose the line-of-sight minimum observation elevation angle to be equal to 10°, 20°, and 30° to represent different conditions. With an increase of barycenter separation distance, the variation of the lunar intervisible spherical cap also increases, shown in Figure 8a. The maximum lunar intervisible spherical cap reaches nearly 45% of the lunar surface area. Figure 8b indicates that when the line-of-sight minimum elevation angle increases by the same amount, the lunar intervisible spherical cap decreases at a slower rate. Thus, the effects of line-of-sight minimum observation elevation angle to the lunar intervisible spherical cap are considerable. However, if the line-of-sight minimum observation elevation angle is constant, the lunar intervisible spherical cap area becomes larger when the Moon moves from the perigee to apogee.

Spatio-Temporal Coverage Performance at Different Positions on the Lunar Surface
In this section, we first introduce the effects of barycenter separation distance and the line-of-sight minimum observation elevation angle of the lunar surface on the lunar intervisible spherical cap. Next, we assume that a Moon-based platform is equipped at different positions on the lunar surface to determine the observation performance from different positions on the Moon. Figure 8 displays the influences of barycenter separation distance and the line-of-sight minimum observation elevation angle on the lunar intervisible spherical cap. As described in [33], in the full observation regions (longitude between 80 • W-80 • E and latitude between 80 • S-80 • N), which is less affected by lunar terrain, the line-of-sight minimum elevation angle should be larger than 20 • . Thus, we choose the line-of-sight minimum observation elevation angle to be equal to 10 • , 20 • , and 30 • to represent different conditions. With an increase of barycenter separation distance, the variation of the lunar intervisible spherical cap also increases, shown in Figure 8a. The maximum lunar intervisible spherical cap reaches nearly 45% of the lunar surface area. Figure 8b indicates that when the line-of-sight minimum elevation angle increases by the same amount, the lunar intervisible spherical cap decreases at a slower rate. Thus, the effects of line-of-sight minimum observation elevation angle to the lunar intervisible spherical cap are considerable. However, if the line-of-sight minimum observation elevation angle is constant, the lunar intervisible spherical cap area becomes larger when the Moon moves from the perigee to apogee.  Figure 9 shows the variations of intervisible spherical cap, starting at 00:00 UTC on 1 January 2010 and ending at 20:00 UTC on the same day at four-hour intervals. Figure 8a-d shows that the invisible area is mainly limited to the edge of the near side of the Moon, and every subfigure shows a few differences. To elucidate the coverage's change in one day, the difference figures are shown in Figure 10. Over time, the invisible area that starts from the edge area gradually extends to the center of the Moon. However, the ratio of the areal change in one day is just 5% of the entire lunar surface, which can be considered negligible. In summary, if a Moon-based sensor is located at a place where it can observe the Earth even when at the edge of the Moon, the sensor is likely to observe Earth in one day.   Figure 9 shows the variations of intervisible spherical cap, starting at 00:00 UTC on 1 January 2010 and ending at 20:00 UTC on the same day at four-hour intervals. Figure 8a-d shows that the invisible area is mainly limited to the edge of the near side of the Moon, and every subfigure shows a few differences. To elucidate the coverage's change in one day, the difference figures are shown in Figure 10. Over time, the invisible area that starts from the edge area gradually extends to the center of the Moon. However, the ratio of the areal change in one day is just 5% of the entire lunar surface, which can be considered negligible. In summary, if a Moon-based sensor is located at a place where it can observe the Earth even when at the edge of the Moon, the sensor is likely to observe Earth in one day.  Figure 9 shows the variations of intervisible spherical cap, starting at 00:00 UTC on 1 January 2010 and ending at 20:00 UTC on the same day at four-hour intervals. Figure 8a-d shows that the invisible area is mainly limited to the edge of the near side of the Moon, and every subfigure shows a few differences. To elucidate the coverage's change in one day, the difference figures are shown in Figure 10. Over time, the invisible area that starts from the edge area gradually extends to the center of the Moon. However, the ratio of the areal change in one day is just 5% of the entire lunar surface, which can be considered negligible. In summary, if a Moon-based sensor is located at a place where it can observe the Earth even when at the edge of the Moon, the sensor is likely to observe Earth in one day.  Therefore, the variations of intervisible spherical cap for one month are calculated as shown in Figure 11. The time interval between adjacent subfigures is five days. Compared with Figure 9, the daily variation is much more distinct, and can be regarded as the movement of the nadir point that first moves northward; half a month later, the nadir point comes back to the lunar equator then moves southward. Therefore, the invisible area becomes larger in the south but disappears in the north on the sixth day. This condition is opposite to that of the 21st day. In detail, the area of lunar intervisible spherical cap is related to the barycenter separation distance, and whether or not the area can be observed on the Earth is relative to the angle between the equatorial plane of the Moon and the line between barycenter of the Earth and the Moon. The result is the foundation for considering lunar polar regions as the Moon-based platform's site. Because Earth polar regions exist permanent illumination region and permanent shadowed area. As for the permanent illumination region, it will be helpful for the energy supplement. For the permanent shadowed area, on the one hand, there may exist ice or water, which is the primary condition for site selection. On the other hand, we do not need to consider the potential damage caused by solar invasion to a Moon-based sensor. Thus, this result can provide the accurate visible time for Earth observations at the lunar edge and the range of lunar visible area, and they are related to Earth-Moon distance and the latitude of the projection of the barycenter of the Earth on the lunar surface, and influenced greatly by the lunar orbital period. Therefore, the variations of intervisible spherical cap for one month are calculated as shown in Figure 11. The time interval between adjacent subfigures is five days. Compared with Figure 9, the daily variation is much more distinct, and can be regarded as the movement of the nadir point that first moves northward; half a month later, the nadir point comes back to the lunar equator then moves southward. Therefore, the invisible area becomes larger in the south but disappears in the north on the sixth day. This condition is opposite to that of the 21st day. In detail, the area of lunar intervisible spherical cap is related to the barycenter separation distance, and whether or not the area can be observed on the Earth is relative to the angle between the equatorial plane of the Moon and the line between barycenter of the Earth and the Moon. The result is the foundation for considering lunar polar regions as the Moon-based platform's site. Because Earth polar regions exist permanent illumination region and permanent shadowed area. As for the permanent illumination region, it will be helpful for the energy supplement. For the permanent shadowed area, on the one hand, there may exist ice or water, which is the primary condition for site selection. On the other hand, we do not need to consider the potential damage caused by solar invasion to a Moon-based sensor. Thus, this result can provide the accurate visible time for Earth observations at the lunar edge and the range of lunar visible area, and they are related to Earth-Moon distance and the latitude of the projection of the barycenter of the Earth on the lunar surface, and influenced greatly by the lunar orbital period.
In this study, based on the characteristics of the Moon-based Earth observation, we built an intervisibility geometry model that, on one hand, can be used to evaluate whether the hemisphere of the Earth can be seen from a certain position on the lunar surface. On the other hand, the area of the lunar intervisible spherical cap is expressed by an analytical expression that is in connection with the line-of-sight minimum observation elevation angle and the barycenter separation distance. From the variation of the barycenter separation distance, we found a suitable time to observe the Earth because the lunar intervisible spherical cap reached the maximum. In this study, based on the characteristics of the Moon-based Earth observation, we built an intervisibility geometry model that, on one hand, can be used to evaluate whether the hemisphere of the Earth can be seen from a certain position on the lunar surface. On the other hand, the area of the lunar intervisible spherical cap is expressed by an analytical expression that is in connection with the line-of-sight minimum observation elevation angle and the barycenter separation distance. From the variation of the barycenter separation distance, we found a suitable time to observe the Earth because the lunar intervisible spherical cap reached the maximum.
In our work, based on an intervisiblility geometry model, an analytical formula of the lunar intervisible spherical cap that can observe the hemisphere of the Earth was derived. The lunar intervisible spherical cap is related to the line-of-sight minimum observation elevation angle and the barycenter separation distance, and the results show that the effects of the line-of-sight minimum observation elevation angle to the lunar intervisible spherical cap are considerable. This contrasts with Ren et al. [20] and Ye et al. [21], whose geometry models do not consider the effects of minimum elevation angle to the spatial coverage, and the positions of their Moon-based sensor satisfy the condition of observing one point on the Earth rather than the entire hemisphere of the Earth. In addition, Wang et al. [34] and Sui et al. [35] also discusses the observation characteristics of a Moonbased platform, but they concentrate on the polar regions and Moon-based sensors located in certain places; thus, the two distinct lines as shown in Figure 7 are not found in their study. Our work focuses on Earth hemispheric observations, including polar regions, and sensors anywhere on the lunar intervisible spherical cap can achieve hemispheric observations. Ye et al. [22] expresses regions on the lunar surface where can achieve hemisphere's observation by fixed latitude and longitude, which is not accurate because the area of intervisible spherical cap is related to the distance between the barycenter of the Earth and the barycenter of the Moon.
In recent years, Europe has been planning to build a lunar research station, as has China [36]. Meanwhile, the United States is preparing for future lunar exploration. The spatio-temporal characteristics are the primary condition for Earth observations. Our work can provide additional insights into spatio-temporal characteristics of a Moon-based platform. Additionally, it can determine the change of area on the Moon where the whole hemisphere of the Earth can be observed, which can provide support for calculating the accurate time for a Moon-based Earth hemispheric observation.

Conclusions
Based on the unique observation geometry of a Moon-based platform and according to the intervisibility geometry model, we derived an analytical formula for the lunar intervisible spherical In our work, based on an intervisiblility geometry model, an analytical formula of the lunar intervisible spherical cap that can observe the hemisphere of the Earth was derived. The lunar intervisible spherical cap is related to the line-of-sight minimum observation elevation angle and the barycenter separation distance, and the results show that the effects of the line-of-sight minimum observation elevation angle to the lunar intervisible spherical cap are considerable. This contrasts with Ren et al. [20] and Ye et al. [21], whose geometry models do not consider the effects of minimum elevation angle to the spatial coverage, and the positions of their Moon-based sensor satisfy the condition of observing one point on the Earth rather than the entire hemisphere of the Earth. In addition, Wang et al. [34] and Sui et al. [35] also discusses the observation characteristics of a Moon-based platform, but they concentrate on the polar regions and Moon-based sensors located in certain places; thus, the two distinct lines as shown in Figure 7 are not found in their study. Our work focuses on Earth hemispheric observations, including polar regions, and sensors anywhere on the lunar intervisible spherical cap can achieve hemispheric observations. Ye et al. [22] expresses regions on the lunar surface where can achieve hemisphere's observation by fixed latitude and longitude, which is not accurate because the area of intervisible spherical cap is related to the distance between the barycenter of the Earth and the barycenter of the Moon.
In recent years, Europe has been planning to build a lunar research station, as has China [36]. Meanwhile, the United States is preparing for future lunar exploration. The spatio-temporal characteristics are the primary condition for Earth observations. Our work can provide additional insights into spatio-temporal characteristics of a Moon-based platform. Additionally, it can determine the change of area on the Moon where the whole hemisphere of the Earth can be observed, which can provide support for calculating the accurate time for a Moon-based Earth hemispheric observation.

Conclusions
Based on the unique observation geometry of a Moon-based platform and according to the intervisibility geometry model, we derived an analytical formula for the lunar intervisible spherical cap to determine suitable places where sensors can be equipped to observe the hemisphere of the Earth, which can thus increase awareness of the spatio-temporal characteristics of a Moon-based platform. The Earth observations show daily periodicity and a 27-day cycle that corresponds to the lunar orbit period that also occurs. For the average global observation duration, two distinct lines occur within the ranges of 61 • N to 71 • N and 61 • S to 71 • S. Regions between the top line and 90 • N or between the bottom line and 90 • S change with the latitude of the nadir point; when the maximum latitude of the nadir point increases, the widths of the two regions become larger. However, the observation durations of regions between two lines are in accordance. The lunar intervisible spherical cap on the lunar surface varies with the line-of-sight minimum elevation angle as well as the barycenter separation distance. With an increase of the line-of-sight minimum elevation angle, the lunar intervisible spherical cap decreased, although the opposite effect was observed for barycenter separation distance. If a Moon-based sensor is located at a place where it can observe the Earth's hemisphere, it is likely to observe the hemisphere within one day, and its small period is also approximately 27 days. From the perspective of the area of the lunar intervisible spherical cap, our study can provide guidance in calculations of accurate time for Earth hemispheric observations from a Moon-based platform.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
Detailed calculation of the percentage of loss of lit area viewed from a Moon-based platform and DSCOVR are expressed below. Figure A1 shows the observed lit spherical cap from a Moon-based platform and DSCOVR. It is not drawn to scale. Because considering the Earth as ellipsoid or not has negligible effects of the visible area, thus, these calculations are in the situation that considers the Earth as a spheroid model. Assuming the radius of the Earth is rE and the distance between a Moon-based platform (DSCOVR) and the nadir point on the Earth is d, thus, the gray or the green spherical cap can be shown as: Thus, the ratio between the observed lit spherical cap and the lit hemisphere can be calculated as: For a Moon-based platform, it is assumed the distance d is equal to 380,000 km. For DSCOVR, it is assumed the distance d is equal to 1,500,000 km. The radius of the Earth is 6371 km. Thus, the percentages of loss of area are shown as: where LM means percentage of loss of area viewed from a Moon-based platform and LD means percentage of loss of area viewed from DSCOVR. It is worth noting that all the calculations are in the assumption that a Moon-based platform can observe the full lit Earth. In the situation that a Moonbased platform cannot observe the full lit Earth, the percentage of loss of area viewed from a Moonbased platform is larger than 0.0165. In general, the minimal percentage of loss of area viewed from a Moon-based platform is 0.0165 and the percentage of loss of area viewed from a Moon-based platform is about 0.0042. Assuming the radius of the Earth is r E and the distance between a Moon-based platform (DSCOVR) and the nadir point on the Earth is d, thus, the gray or the green spherical cap can be shown as: Thus, the ratio between the observed lit spherical cap and the lit hemisphere can be calculated as: For a Moon-based platform, it is assumed the distance d is equal to 380,000 km. For DSCOVR, it is assumed the distance d is equal to 1,500,000 km. The radius of the Earth is 6371 km. Thus, the percentages of loss of area are shown as: where L M means percentage of loss of area viewed from a Moon-based platform and L D means percentage of loss of area viewed from DSCOVR. It is worth noting that all the calculations are in the assumption that a Moon-based platform can observe the full lit Earth. In the situation that a Moon-based platform cannot observe the full lit Earth, the percentage of loss of area viewed from a Moon-based platform is larger than 0.0165. In general, the minimal percentage of loss of area viewed from a Moon-based platform is 0.0165 and the percentage of loss of area viewed from a Moon-based platform is about 0.0042.