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. 2023 Sep 13;9(37):eadh4302. doi: 10.1126/sciadv.adh4302

Past extent of lunar permanently shadowed areas

Norbert Schörghofer 1,†,*, Raluca Rufu 2
PMCID: PMC10499310  PMID: 37703378

Abstract

As the Moon migrated away from Earth, it experienced a major spin axis reorientation. Permanently shadowed regions (PSRs), which are thought to have trapped ices and are a main focus of lunar exploration, appeared and grew after this (Cassini state) transition and are often younger than their host craters. Here, we calculate the lunar spin axis orientation and the extent of PSRs based on recent advances for the time evolution of the Earth-Moon distance. The solar declination reached twice its current value 2.1 billion years (Ga) ago, when the PSR area was about half as large. The PSR area becomes negligible beyond 3.4 Ga ago. The site of an artificial impact in Cabeus Crater, where various volatiles have been detected, became continuously shadowed only about 0.9 Ga ago, and hence, cold-trapping has continued into this relatively recent time period. Overall estimates for the amount of cold-trapped ices have to be revised downward.

Most of the Moon's permanently shadowed areas arose less than 2.2 billion years ago and some trapped ice during the recent past.

INTRODUCTION

The current small lunar obliquity (ɛ = 6.7°, the angle between the spin axis and the orbit normal) together with a comparable orbital inclination (i = 5.1° with respect to the normal to the ecliptic plane) results in a small maximum solar declination (θ = 1.5°, the maximum elevation of the Sun above the lunar poles). This configuration allows for permanently shadowed regions (PSRs) at the poles, where, because of low temperature, ices can accumulate (1, 2). The lunar axis experienced a major reorientation when the Moon transitioned from Cassini state 1 to 2, which occurred at ⁓34 Earth radii (3), and the axis tilt may have briefly reached 77° before it was damped to small values (3, 4). Such high obliquities must have resulted in the loss of all ice deposits. PSRs appeared and then grew after this transition and are often younger than their host craters (1, 5). The time evolution of the Moon-Earth distance remained a conundrum for half a century, as it is difficult to assign Earth-Moon distances to absolute times in a way that is consistent with available constraints (6). Recently, Farhat et al. (7) found a solution that agrees with geological proxies for the history of the Earth-Moon system. This breakthrough enables us to calculate the lunar obliquity and the extent of PSRs as a function of time. Impacts and outgassing are potential sources of water but peaked early in lunar history, so the age of PSRs is a dominant factor for the amount of water ice trapped in the lunar polar regions, which is the prime target of upcoming crewed and uncrewed missions to the Moon.

RESULTS

Tides raised on the Moon by Earth despun the Moon to a synchronous rotation and its spin orientation to a Cassini state (a configuration in which the rotational axis, lunar orbital normal, and the normal to the Laplacian plane are all coplanar) (3). The tidal decay timescale for the spin axis to reach a Cassini state is small compared to the tidal evolution. Hence, here we assume that the lunar spin axis follows the stable Cassini state 2 (the current tidally locked state), equation 1 in (3). We use values for Earth’s obliquity and distance, as provided by Farhat et al. (7), and current moment of inertia values (8). The orbital inclination i of the Moon was obtained using a tidal model for planetary tides (9) and agrees with previous results (10). We combined these quantities to obtain the maximum solar declination θ (currently 1.5°) as a function of time, where θ is the difference between the lunar obliquity and the orbital inclination.

Figure 1A shows the Earth-Moon distance according to Farhat et al. (7). The Moon initially moved outward rapidly and experienced the Cassini state transition about 4.1 billion years (Ga) ago. Figure 1B shows the lunar obliquity ɛ and the declination θ over the past 3.5 Ga. The declination θ was 3°, twice the current value, about 2.1 Ga ago; θ = 2° was reached 0.61 Ga ago.

Fig. 1. Evolution of Moon distance and axis orientation.

Fig. 1.

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Time dependence of (A) the Earth-Moon distance according to Farhat et al. (7) and (B) the lunar obliquity ɛ and maximum solar declination θ = ɛ − i, where i is the orbital inclination. Lunar geologic periods are shown at the top of (A).

We use ray tracing to determine the extent of PSRs based on a LOLA (Lunar Orbiter Laser Altimeter) shape model (see Methods). To account for the size of the solar disk, 1/4° was added to the declination. The present-day PSR area agrees with previous results (Methods) (11). Figure 2 shows the resulting PSR maps for three solar declinations. PSRs formed relatively early in the craters Shackleton, de Gerlache, and Idel’son L, among others, but emerged more recently in Sverdrup and Amundsen. The PSRs within the “Three Amigos” (Haworth, Shoemaker, and Faustini) share similar ages. The maps of past PSRs are archived online for maximum solar declinations ranging from 1.5° to 10° (see Acknowledgments).

Fig. 2. Past and current PSRs in the lunar polar regions.

Fig. 2.

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Colored patches show the extent of PSRs for maximum solar declinations of 1.5° (blue, close to current), 3° (green, 2.1 Ga ago), and 6° (red, 3.3 Ga ago) in polar stereographic projection for the south and north polar regions with present-day topography. The background grayscale map is maximum direct solar irradiance for the present day.

Figure 3 shows the PSR area in each polar region as a function of time. The PSR area here is measured poleward of 80° latitude. Many PSRs are found farther equatorward but are often not cold traps because of terrain irradiance (12). (Cold traps are defined by surface temperatures of negligible sublimation rate, ≲110 K, and lie within PSRs.) The PSR area and the cold trap area are more closely related poleward of 80° than they are globally. The cumulative PSR area was half as large 2.2 Ga ago. It reached a quarter of its current size ∼3.1 Ga ago (assuming that all craters already existed at that time). The PSR area becomes negligible beyond 3.4 Ga ago. The cold trap area is expected to have decreased more quickly than the PSR area because terrain irradiance increases with sun elevation (5, 13). Using the graph in Fig. 3 to integrate time with respect to area provides an estimate for the average age of a PSR of 1.8 Ga. This is an upper bound because the craters themselves can have formed more recently.

Fig. 3. PSR area as a function of time assuming present-day topography at 240 m per pixel.

Fig. 3.

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The numbers above the data points indicate the maximum solar declination (of the center of the solar disk) θ.

Small PSRs reside on the floors of some deep craters for solar declinations well above 6° (5). However, craters with a large depth-to-diameter ratio are subject to a large terrain irradiance contribution (13) and therefore less likely to act as cold traps. Shackleton Crater (14), at the south pole, a candidate landing site for the Artemis program, could host an exceptionally old cold trap (5); the crater is estimated to be 3.4 Ga old (15).

Of special interest is the age of the cold trap at the LCROSS (Lunar Crater Observation and Sensing Satellite) impact site in Cabeus Crater, where water ice and other volatiles were detected (16). Figure 4 shows the extent of past PSRs at this site. The PSR at the LCROSS impact site disappeared at θ ≈ 2.2°, about 0.9 Ga ago. This constrains the emplacement age of the ice because volatiles could have only accumulated after the PSR formed.

Fig. 4. Extent of past PSRs in Cabeus Crater based on a 40m LOLA DEM.

Fig. 4.

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The location of the LCROSS impact is marked with a bold cross and became continuously shadowed at θ ≈ 2.2° (about 0.9 Ga ago). Color contours are the boundaries of PSRs and are labeled with values of θ. Gray contours are topography relative to a sphere in 100-m height intervals. X and Y are coordinates in polar stereographic projection.

DISCUSSION

Uncertainties in axis orientation

The uncertainties in the Earth-Moon distance provided by Farhat et al. (7) are small for the most recent 3.1 Ga (Fig. 1A). Over the past 3 Ga, they translate into uncertainties of less than 10−4° in obliquity. The inclination of the lunar orbit could have been altered by close encounters of large bodies with the Earth-Moon system. Over the past 3 Ga, close encounters likely did not involve bodies large enough to cause substantial changes in the lunar inclination (see Methods). These inclination changes translate into less than 0.04° changes in obliquity and 0.02° in maximum solar declination. For example, θ = 2.2 ± 0.1° corresponds to an age of 0.93 ± 0.13 Ga for the PSR at the LCROSS location in Cabeus Crater (Fig. 4). Encounters with 1000-km-size bodies may have occurred before 3 Ga ago and can cause changes of ∼0.2° to the lunar inclination.

Changes in the location of the pole relative to the surface are known as true polar wander. They can occur because of changes in the moments of inertia caused by large impacts or mantle convection. True polar wander due to impacts is estimated to be less than 2° over the past 3.8 Ga (17), but the largest impacts occurred early in the Moon’s history (18), and the uncertainties shrink rapidly thereafter (see Methods). Mantle convection due to radiogenic heating has been discussed as a source of true polar wander (19, 20). Hypothetical model scenarios span a wide range of possibilities (20), and the history of PSRs could have been altered to an extent that is difficult to ascertain.

Implications for the abundance of ice

Large (≳50 km) craters have model ages of about 4 Ga (21, 22) and therefore are older than the PSRs they host. A fraction of water molecules that originate anywhere on the surface can be captured by polar cold traps after lateral transport by an exosphere or a transient atmosphere. Most sources of lunar water predate most present-day cold traps, and previous estimates for the expected amount of ice in cold traps have to be revised downward dramatically. In the statistical mean of scenarios considered by Cannon et al. (23), impacts and outgassing delivered 24 times more ice to the polar regions in the period 4.25 to 3.75 Ga than in the 3.75 Ga since then. Delivery of comets and hydrated asteroids peaked early (23). The Imbrian period (3.85 to 3.20 Ga ago) barely had any PSRs. Volcanic outgassing peaked 3.7 to 3.5 Ga ago but continued to at least 2 Ga ago (24–27); thus, late-stage outgassing overlapped with early PSRs. Solar wind–generated water (28–30) may be a relatively important component of the water ice still preserved. The fact that the PSR at the LCROSS impact site is only 0.9 Ga old suggests that volatiles continued to be delivered and cold-trapped in the Copernican period (1.1 Ga to present).

Maps of old PSRs (Fig. 2) can serve as guide for the search of ice deposits that will soon be conducted by landed missions to the Moon. The longevity of relic buried ice outside of cold traps (“subsurface stability”) was not evaluated here, but in the absence of an ancient ice reservoir, this requires that the ice was rapidly covered with a protective layer after it was deposited, which could only have taken place at a small fraction of the area with subsurface stability.

Mercury entered its current Cassini state sometime early in its history (31, 32), and its PSRs are most likely older than the Moon’s. The younger age of the lunar PSRs potentially explains why Mercury’s cold traps contain far more ice than the Moon’s (2).

METHODS

Terrain shadowing calculations

We use ray tracing to determine the height of horizons based on a LOLA shape model at 240 m per pixel resolution (33) and a triangulated mesh. Intel Embree ray tracing kernels were used in conjunction with a set of programs developed for terrain shadowing calculations on airless bodies (see Acknowledgments) (34). We assume a circular orbit at 1 astronomical unit and calculate shadows for 360 azimuths over a solar day at solstice. To account for the size of the solar disk, 1/4° was added to the declination, and PSRs are defined as areas where the direct solar irradiance is always zero. The present-day PSR area agrees with previous results for the same spatial resolution (11), specifically 10,921 versus 10,894 km2 poleward of 82.5°N and 13,589 versus 13,217 km2 poleward of 82.5°S. Our declination was exactly 1.5°, slightly lower than that for the current lunar orbit used in (11), which may account for the remaining differences.

Uncertainties in past lunar inclination

The solar declination is a function of the lunar inclination and obliquity, θ = ɛ − i. The lunar inclination may slightly vary from the simple tidal-driven evolution assumed in this work due to collisionless encounters with Earth-crossing planetesimals (35). The damped Cassini state lunar obliquity depends on the Earth-Moon distance, Earth’s obliquity, and lunar inclination. The uncertainties provided by Farhat et al. (7) on the Earth-Moon distance and Earth’s obliquity lead to only small uncertainties in the damped lunar obliquity compared to the errors invoked by the inclination uncertainties. We estimate below the possible inclination variations and how these translate to the solar declination uncertainty.

The encountering population in the past 3.5 Ga is largely composed of the asteroid population and near-Earth asteroids (NEAs) (18). To estimate the number of collisionless encounters, we use the cumulative number of impacts onto Earth (18) and multiply them by the Hill sphere cross-sectional area compared to Earth’s cross section. The flux accounts for the decline of asteroids during early epochs, while the NEAs are represented by a constant flux in the past 3 Ga. The fluxes are normalized by the current size distribution of the asteroid belt and NEAs (36, 37), where we consider only bodies with diameters d > 10 km.

To estimate the maximum lunar inclination variation from one planetesimal encounter, we performed 10,000 scattering simulations between Earth, Moon, and an encountering small body. We use REBOUND N-body code with a 15th-order Gauss-Radau integrator (IAS15) (38, 39) with adaptive time step, which allows for collision detection. In each simulation, we place Earth and an additional body of diameter d = 10, 50, 100, or 1000 km (assumed density of 3 g/cm3) at the closest approach with a random distance between R⊕ and RHill (where R⊕ and RHill are Earth radius and Hill radius, respectively) and a random velocity direction where r→⋅V→>0 (where r→ and V→ are the relative position and velocity of the body from Earth). The velocity is set to V2 =GM⊕/r+V∞2, where V∞ = 16.4 km/s (36), G is the gravitational constant, and M⊕ is Earth’s mass. We integrate backward in time until the distance between Earth and the encountering body is RHill. We then place the Moon on a circular coplanar orbit at a semimajor axis of 60R⊕ (close to the current value, ensuring the maximum torque on the lunar orbit) and integrate forward until the body exits Earth’s Hill sphere and check the lunar inclination after each encounter. We find that none of the d = 10 km encounters changed the lunar inclination above the numerical significance (Δimax,10 km < 10−6°), while the maximum inclination changes recorded were Δimax,50 km = 3 × 10−5°, Δimax,100 km = 2 × 10−4°, and Δimax,1000 km = 0.24° for the D = 50 km, d = 100 km, and d = 1000 km, respectively.

It is not certain how many D > 500-km bodies existed at each time (currently, there are only three in the main belt and no NEAs of that size). Assuming that the size distribution has not changed, there have probably been only a few close encounters with this population ∼3.5 to 3 Ga ago (with the cumulative number of encounters <1 after 3 Ga ago). Because of the small number of events and uncertainties, we assume a 0.24° uncertainty to times before 3 Ga, which is the maximum inclination change recorded in our scattering simulations with a 1000-km body.

The lunar orbit will experience a random walk-like behavior given N(d > 10 km) encountering bodies. We estimate that the maximum lunar inclination due to the collisionless encounters is N(D>10km)10−6+N(D>50km)Δimax,50km+N(D>100km)Δimax,100km. This is an overestimation of the probable inclination variation with time because most encounters would result in a considerably smaller inclination change than the recorded maximum (e.g., 70% of the encounters with the 100-km body resulted in <10−6° change). The resulting inclination uncertainty for times <3 Ga is <0.022°, invoking an uncertainty of 0.015° on the solar declination (note that a larger inclination invokes a larger obliquity; hence, the resulting solar declination is slightly smaller than the inclination uncertainty).

Uncertainties due to true polar wander caused by impacts

The online data accompanying Smith et al. (17) provide 100 stochastic time series for the true polar wander caused by impacts (https://pgda.gsfc.nasa.gov/products/86). The co-latitudes for the stochastic impact histories were interpolated to specific times in intervals of 0.1 Ga (Fig. 5). The co-latitude represents the distance between the past and current spin pole and thus the uncertainty in past pole orientation due to mass redistributions caused by impacts. By 3.2 Ga, the mean co-latitude fell below 0.2°. By 0.9 Ga ago, the pole likely differed by less than 0.1° and probably only by 0.04°.

Fig. 5. Co-latitude of the lunar spin axis due to the true polar wander caused by impacts.

Fig. 5.

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The mean and maximum co-latitudes among 100 histories studied in (17) are shown as a function of time.

Acknowledgments

We thank O. Aharonson, S. Bertone, S. Marchi, E. Mazarico, D. Nesvorný, S. Potter, and L. Rubanenko for insightful discussions.

Funding: This material is based on work supported by the National Aeronautics and Space Administration under grant no. 80NSSC22K1339 issued through the Lunar Data Analysis Program, through the Solar System Exploration Research Virtual Institute (SSERVI) Cooperative Agreement 80NSSC19M0216 (GEODES), and by a Sagan Fellowship to R.R.

Author contributions: R.R. performed the obliquity and inclination integrations. N.S. conceptualized the study and performed the PSR calculations. Both authors contributed to the writing of the manuscript.

Competing interests: The authors declare that they have no competing interests.

Data and materials availability: The results for the Earth-Moon distance (7), which served as input to our calculations, are available in electronic form at https://astrogeo.eu/?page_id=553. Obliquity and orbital inclination as a function of time, which are an output of our calculations, are available in electronic form at https://github.com/ralucaru/Lunar-Obliquity. The LOLA shape model (LRO-L-LOLA-4-GDR-V1.0) was downloaded from http://imbrium.mit.edu/BROWSE/LOLA_GDR/. The Python programs are at https://github.com/sampotter/python-flux interface with a wrapper for the Embree ray-tracer (https://github.com/sampotter/python-embree, https://www.embree.org/). Our PSR maps for various solar declinations are posted at https://doi.org/10.5281/zenodo.7549269. All data needed to evaluate the conclusions in the paper are present in the paper.

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