The scientific benefits of observations from the Moon for the Earth, exoplanet, and astrophysics studies are discussed in several recent papers (Marshak et al., 2020; Gorkavyi et al., 2021; Boyd et al., 2022). Although current Earth-observing satellites can produce high-resolution images, low Earth orbit (LEO) sensors can only scan a small portion of the globe at a given time, while geosynchronous equatorial orbit (GEO) sensors can provide temporally continuous, though lower-resolution, observations of a significant, but fixed, portion of the Earth's disk. The Earth Polychromatic Imaging Camera (EPIC) on the Deep Space Climate Observatory (DSCOVR) clearly stands apart, observing the entire Sun-illuminated Earth from the L1 Sun–Earth Lagrange point (Marshak et al., 2018). The L1 location, however, limits phase (i.e., Sun–Earth–camera) angles to a nearly backscattering direction (up to ∼ 178∘). The phase angle interval from 2 to 12∘ is determined by the trajectory of the DSCOVR space observatory, which does not rest at the Lagrange point but moves around it. It is too “noisy” to transmit data directly from the Earth–Sun line; thus, the phase angle is bigger than 2∘. A compact, lightweight, autonomous camera and a spectrometer on the Moon's surface offer a unique opportunity to complement these observations and to image the full range of Earth's phases, potentially advancing Earth science in many ways (Marshak et al., 2020; Gorkavyi et al., 2021) – some examples are listed as follows:

observing ocean–cloud glints at different phase angles;

The first telescopic image of the Earth from the Moon was obtained during the expedition of Apollo 16 in 1972 using an ultraviolet telescope (Carruthers and Page, 1972) – see Fig. 1. In later years, prospects for lunar observations of the Earth have been discussed in many papers (e.g., Foing, 1996; Moccia and Renga, 2010). Observations of the Earth with instruments mounted on the Moon have been actively discussed in recent years (Hamill, 2016). Impressive prospects for observing the Earth from the Moon in the visible spectrum are demonstrated in the pictures taken by the Lunar Reconnaissance Orbiter (LRO) in 2015. The difference in distances (Earth–Moon and Earth–L1) leads to the fact that a telescope with the same field of view sees the Earth with different angular sizes and resolutions (Fig. 2).

One of the objectives of the Chinese space program is Moon-based observation of the Earth (Li et al., 2019; Guo et al, 2019). A lunar lander, Chang'e-3 (which landed on the Moon in 2014 and is still working), is equipped with a 5 cm ultraviolet telescope and extreme UV camera and studies changes in the Earth's plasmasphere in the UV range (He et al., 2016).

One of the tasks facing the US Artemis program is as follows:

Use the Moon as a platform for Earth-observing studies … The observations from the Moon will have higher resolution than would similar observations made from L1. Myriad science investigations targeting topics such as lightning, Earth's albedo, atmosphere, and exosphere …, the oceans, infrared emission, and radar interferometry may be accomplished from the surface of the Moon. The Moon also offers a unique vantage point for full-disk observations … (Artemis III Science, 2020).

Because of tidal locking, the Moon's rotation around its axis is synchronized with its orbital rotation around the Earth. Therefore, the Moon always faces the Earth on one side, and the task of observing the Earth from the Moon seems simple: the Earth must hang motionless in the lunar sky, rotating around its axis. In reality, the Earth moves along a complex trajectory in the sky of the Moon due to lunar librations in latitude and longitude. On the one hand, the librations of the Moon cause the Earth to shift from the field of view of the lunar telescope, which forces one to turn the telescope to track the Earth's observable movement in the Moon's sky (guiding); on the other hand, for a fixed-slit spectrometer, the librations of the Moon can be useful because they make the Earth move through the fixed field of view of the instrument (across the slit). These librations can serve as a natural mechanism for scanning the Earth when observed from the Moon, which allows for the use of slit spectrometers and telescopes on a fixed platform. This paper takes into account lunar librations and analyzes the conditions for observing the Earth from the Moon.

Observations of the Earth from the Moon require accounting for the geometry of the relative position of the centers of the Earth and the Moon, the inclinations of their axes, and the libration effects (Meeus, 1991, 2000; Guo et al., 2018; Xu and Chen, 2019; Huang et al., 2020).

Libration of the Moon in latitude. The angle between the Moon's axis of rotation (N–S or north–south) and the normal to the plane of its orbit around Earth (PP′) is 6.68∘ (Fig. 3). This causes the libration of the Moon in latitude with the same amplitude and with a period of draconic month TD = 27.21222 d or 653.0933 h (the interval between consecutive passages of the Moon through the same node of the orbit; an orbital node is either of the two points where a lunar orbit intersects an ecliptic plane to which it is inclined) – see, for example, Meeus (1991, 2000). As a result of this inclination, parts of the Moon's polar regions are accessible for observations from the Earth.

Libration of the Moon in longitude. The Moon moves around the Earth in an elliptical orbit with an average eccentricity (or deviation of an orbit from circularity) e=0.055 (it varies between 0.0255 and 0.0775) and a period of anomalistic month TA = 27.55455 d or 661.3092 h – the interval between consecutive passages of the Moon through the perigee rmin⁡=a(1-e) or the apogee rmax⁡=a(1+e) of its orbit, where a is the semi-major axis a = 384 399 km (Fig. 4). This causes libration in longitude with an amplitude of 7.9∘ (see, for example, Meeus (1991, 2000)).

The longitudinal libration consists of two components:

An ellipse with a small eccentricity (in the first approximation) can be described as a circle around the Earth, which is shifted from the center of the circle (point O) by the distance OE = ea (see Fig. 4). This displacement leads to the fact that the observer from the Earth begins to see part of the lateral surfaces of the Moon (Fig. 4).

Figure 5 shows the librations of the Moon in longitude and latitude for 2022 (Espenak, 2021). The selenographic coordinate system repeats the Earth's; therefore, the selenographic center of the Moon's disk is the intersection point of the lunar equator and the lunar prime meridian. The selenographic zero corresponds to the average position of the center of the visible disk of the Moon. At any particular moment in time, the center of the visible disk of the Moon can shift from the selenographic zero due to libration. If the apparent center shifts along the lunar equator, then we call this shift the relative longitude of the libration; if it shifts along the meridian, then we call this shift the relative latitude of the libration. In other words, latitude and longitude libration (or relative libration) is the visual displacement of the selenographic center of the Moon's disk (0∘ lunar latitude and 0∘ lunar longitude) relative to the center of the visible disk of the Moon. Libration in longitude correlates with variations in the Earth–Moon distance and changes more strongly with time than libration in latitude.

The libration of the Moon discussed above (Figs. 3 and 4) is called optical libration. The tidal action of the Earth causes physical libration associated with a change in the period of the Moon's own rotation. Physical libration is only 2 arcmin – that is, much less than optical libration. In the calculations (Fig. 5), physical libration is taken into account along with optical libration.

Obviously, the discussed librations of the Moon are directly related to the observation point on the Moon's surface: if the observation point on the Moon changes its angle relative to the Earth–Moon line, then the Earth also changes its position in the lunar sky by the same amount but the different sign (in an approximation where the size of the Moon can be neglected compared to the Moon–Earth distance). In other words, if the lunar telescope raises its line of sight by +5∘ above the line connecting the centers of the Earth and the Moon, then the Earth goes down from the telescope's line of sight by -5∘.

An observer can see the Earth from any point in the Moon's visible hemisphere (Fig. 6). The location of the observer will affect (i) the position of the zero point of Earth's libration in latitude and longitude and (ii) the orientation of the trajectory of the apparent libration of the Earth in the sky of the Moon.

From the point of view of an Earth observer, lunar librations in latitude and longitude are measured as a relative displacement from the lunar zero longitude and longitude – that is, from the point of the lunar disk taken as the zero point and located in the center of the visible disk of the Moon, near the crater Möstig A. From the point of view of a lunar observer located near this crater at the intersection of the lunar equator and the lunar zero meridian (see point O in Fig. 6), the Earth hangs above a given point on the lunar surface (at the zenith). Therefore, if the observer moves away from this point along the lunar meridian, for example, to the lunar north pole (point N in Fig. 6), then the apparent position of the center of the Earth will also shift, moving to the horizon. When the observer is at the lunar pole, the Earth will hang on the horizon. If the observer goes again to the equator and not along the zero meridian but rather along the meridian with a longitude of 90∘ (for example, to the west; see points NW and W in Fig. 6) – that is, along the border between the visible and invisible hemispheres of the Moon, then the Earth will remain hanging above the lunar horizon but will change the apparent tilt of its axis of rotation. For an observer at the Moon's equator (points W, O, E in Fig. 6), the Earth's axis will tilt 90∘ – that is, the Earth will “lay on its side”.

When the observer reaches the south pole (point S in Fig. 6), the Earth will be turned 180∘ relative to it. Similar changes will occur with the trajectories of the Earth in the sky of the Moon. Libration of the Moon sets the trajectory of the Earth in the lunar sky described by relative latitude and longitude (relative to the point of zero libration, marked with a black dot in Fig. 7). The shape of this trajectory (see the red trajectories in Fig. 7) is strictly defined and does not depend on the position of the observer on the Moon's surface. But the height of the point of zero libration above the horizon depends on the position of the lunar observer, as well as the orientation of the libration trajectory – that is, the rotation of the visible libration trajectory around this point of zero libration. An analogy is a picture hanging on the wall of a room. The pattern in the picture does not depend on the position of the observer, but they can stand on their head and completely change the orientation of the pattern relative to their field of vision.

We can take the latitude and longitude of the libration of the Moon (Giesen, 2018; Espenak, 2021; Fig. 5) and plot the positions of the Earth in the sky of the Moon for each day (Fig. 7). Each dot in the Fig. 7a, b, and c represents the latitude and longitude for a particular day.

Figure 7a shows the visual position of the Earth for the first half of 2022. Figure 7b shows the apparent libration of the Earth for the first half of 2023, and Fig. 7c shows the apparent libration for 4 months (July–September) of 2024. Figure 7d shows the trajectory of the Earth in the sky of the Moon for 3 years (2022–2024). The orientation of the libration pattern in Fig. 7 corresponds to the position of the observer on the line of the zero-meridian S–O. At point O, the zero-libration point is above the observer's head, and when the observer moves to the south pole (point S), the zero-libration point shifts to the horizon.

It can be seen that the shapes of the curves along which the Earth moves in the sky of the Moon change noticeably over 3 years.

The movement of the Moon around the Earth can be characterized by three periods:

draconic month TD = 27.21222 d or 653.0933 h (the period of movement relative to the ascending or descending node)

A beat is an interference pattern between two slightly different frequencies, perceived as a periodic variation in amplitude whose rate is the difference of the two frequencies. As a result of three slightly different lunar periods, we have three different beats or precession frequencies.

The apsidal precession period is TSA=8.85 years and is found by the following formula: 11TSA=1TS-1TA. The nodal precession period is TDS=18.6 years and is found by the following formula: 21TDS=1TD-1TS. The librations of the Moon in latitude and longitude follow a 6-year cycle, when the major axis of the lunar orbit has performed one complete revolution with respect to the line of nodes (Meeus, 1991, 2000; Giesen, 2018): 31TDA=1TD-1TA, with a period of TDA=2190.34 d or 6 (more precisely, 5.99667) anomalistic years (365.259636 d each). All three periods of precessions are connected: 416.00=118.6+18.85. Figure 8a shows the position of the center of the Earth in the lunar sky for 6 years (2022–2027). Figure 8b shows the statistics of the distribution of the 2191 positions of the center of the Earth for this period. The average distribution density of the center of the Earth in squares 1∘ × 1∘ (or the number of entries of the center of the Earth into this square for 6 years) is N‾ = 2191/255 = 8.6 per pixel; in reality N ranges from 0 to 34.

Spectrometric observations through the slit are a common practice for many satellite observations of the Earth. Scanning of the Earth's surface is usually carried out by movement of the low orbit satellites. For observation from the Moon, it is logical to consider the option when scanning occurs due to the libration and diurnal motion of the Earth. The scientific goals for the slit observation are close in both cases.

The angular velocity of a point on the Earth's surface in the field of view of the sensor is caused by two comparable factors: the rotation of the Earth around its axis and lunar libration, which causes a shift in the center of the Earth. The rotation of the Earth around its axis is a well-studied process, but librations of the center of the Earth in the lunar sky are poorly understood and raise many questions. When observing the Earth through the slit of the spectrometer, it will be necessary to take into account both the displacement of the center of the Earth and the Earth's rotation.

The librational apparent motion of the Earth must be taken into account when observing from the Moon and can also become a natural substitute for scanning (Fig. 9). It is proposed that fixed-mount instruments be installed on the Moon's surface, directed towards the Earth.

A hyperspectral sensor (UV, Vis, NIR, IR) will observe Earth passing through fixed vertical slits. The librations of the Moon and the daily rotation of the Earth will serve as a natural scanning mechanism for this spectrometer. This multi-slit spectrometer can be similar to the six-slit hyperspectral limb profiler (LP) on the OMPS aboard Suomi National Polar-orbiting Partnership (S-NPP) LEO satellite, as well as the single-slit hyperspectral Ozone Monitoring Instrument (OMI) on the NASA Earth Observing System (EOS) Aura satellite. Each LP slit uses approximately 1/6 of the detector matrix. A multi-slit spectrometer for observing the Earth from the surface of the Moon can have six to eight slits, whose fields of view are shifted by 2.5∘. Since the maximum angular size of the Earth is 2∘, the angular distance between the lines of sight of neighboring slits must be greater than the angular diameter of the Earth so that light from the Earth does not hit two slits at the same time. Each slit can use the entire matrix because they scan the Earth at different times in turn and do not interfere with each other.

It should be noted that the longitude of the observation point on the Moon affects the orientation of the visible trajectory of the Earth in the sky of the Moon. For example, the diagonally elongated trajectory of the Earth for July–October 2024 (Fig. 7c, for the case of lunar longitudes near 0∘ and for the observer at point S in Fig. 6) will have a different orientation when observed from the zone of lunar longitudes of about 45∘ N for the observer at point NW in Fig. 6. The orientation of the Earth's libration trajectories in the sky of the Moon will change accordingly (see Fig. 10). The sun in the region of the lunar poles moves almost parallel to the horizon, and in the region of the lunar equator, it passes through the zenith, descending vertically to the horizon or rising from it.

Figure 6 shows the orientation of the Earth's crescent from the point of view of different observers on the Moon and helps interpret the orientation of the libration pattern in Fig. 10.

The portion of the illuminated Earth does not change with the position of the Moon's observer, but it changes during the lunar month (Fig. 10, based on data by Espenak, 2021).

These arguments must be taken into account when planning observations of the Earth from the Moon (or when communicating between the Moon and the Earth). Satellites located at the Earth–Moon Lagrange points will move along similar trajectories in the sky of the Moon.

The principal design of the multi-slit spectrometer is shown in Fig. 11. Its main feature is that it uses only one matrix detector for many slits. This is due to the fact that a local object such as the Earth can pass only one slit at a given moment. Therefore, it is possible to image the light from all slits onto a single matrix without compromising observations, although the problem of scattered light may exist and should be studied in the development of a specific instrument. Each slit is directed to a unique position of the Earth in the Moon's sky, but the spectral range of all slits is the same. If the spectra are taken with a slit that occupies a length of 4000 pixels on the detector matrix, then the spectra will be determined from the part of the Earth with a size of ∼ 30 km along the slit. The effective width of this pixel across the slit (i.e., spatial resolution) will depend on the frequency of observations, the width of the slit, and the velocity of the Earth moving across the slit.

The angular diameter of the Earth in the sky of the Moon is 1.8–2.0∘. The typical rate of displacement of the center of the Earth is 1–2∘ per day (see Figs. 7, 10). Therefore, the passage of the Earth through each individual slit of the spectrometer will take 1–2 d. During this time, the Earth makes one to two rotations around its axis, which will allow each slit to receive at least one scan of the entire Earth's surface in one pass. The potential scan frequency depends on the field of view of the device and on the detector matrix used so that the spatial pixel across the slit is comparable to the size of the spatial pixel along the slit. For a detector matrix with a size of 1000–4000 pixels and a field of view of 5 to 15∘, the scan frequency should be 10–100 s.

Important Earth science goals for such a spectrometer are to complement and improve the current DSCOVR-EPIC whole-Earth imaging (Gorkavyi et al., 2021). The acquired data will enable the estimation of aerosol and cloud scattering phase functions, the amount of trace gases, and the surface bidirectional reflectance factor (BRDF).

Due to lunar libration, the center of the Earth for an observer on the Moon moves in a rectangular area with dimensions of 13.4∘ × 15.8∘. The density of the location of the Earth in this rectangular area is an average of 8.6 per square degree over 6 years (2191 d). The density for different parts of the area varies from 0 to 34.

The lunar environment may present serious problems for the operation of sensors due to dust settling and impact on moving parts and due to the influence of high-energy particles and meteoroids. The discussed design of instruments on a fixed mount, with no movement of the external parts of the instruments, significantly reduces the dependence of observations on lunar dust. During the use of such instruments on the lunar surface, the rate of dust settling and the degree of degradation of instruments due to radiation will be clarified, which will make it possible to optimize the design of future instruments and to protect them as much as possible from a hostile environment.

There is a particular interest in observations of the lunar dust environment during total solar and lunar eclipses (which happen ∼ two times a year when the Earth blocks the Sun): the lunar dust glows from scattered sunlight under forward-scattering conditions without saturation of the detector.

The proximity to the Earth (versus the L1 point) and wide variations in phase angle accessible by a Moon-based camera offer unique advantages for observations of the bidirectional land surface reflectance; ocean–cloud glint reflection; whole-globe monitoring of transient volcanic and/or aerosol clouds, as well as polar mesospheric and stratospheric clouds; vegetation; and the twilight zone and shadowed parts of the Earth illuminated by the Moon.