Slope Stability Analysis and Soil Mechanical Properties of Impact Craters around the Lunar South Pole

Abstract

Water ice has been found in the permanently shadowed regions of impact craters around the lunar South Pole, which makes them ideal areas for in situ exploration missions. However, near the rim of impact craters, construction and exploration activities may cause slope instability. As a result, a better understanding of the shear strength of lunar soil under higher stress conditions is required. This paper mainly uses the finite element method to analyze slope stability to determine the position and shape of the slip surface and assess the safety factor. The height and gradient of the slope, the shear strength of lunar soil, and the lunar surface mission all influence the stability of the slope. We also analyze the soil mechanical properties of a soil slope adjacent to the traverse path of the Chang’E-4 Yutu-2 rover. Determining the stability of the slope at the lunar South Pole impact crater under various loading conditions will enhance the implementation of the lunar surface construction program. In this respect, this paper simulates a lunar mission landing at the Shackleton and Shoemaker craters and indicates that areas with higher cohesion lunar soil may be more stable for exploration in the more complex terrain of the South Pole.

Full text

Citation: Huang, Y.; Zhang, J.; Li, B.;
Chen, S. Slope Stability Analysis and
Soil Mechanical Properties of Impact
Craters around the Lunar South Pole.
Remote Sens. 2024,16, 371. https://
doi.org/10.3390/rs16020371
Academic Editor: Christian Wöhler
Received: 5 December 2023
Revised: 6 January 2024
Accepted: 11 January 2024
Published: 17 January 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
remote sensing
Article
Slope Stability Analysis and Soil Mechanical Properties of
Impact Craters around the Lunar South Pole
Yantong Huang 1,2, Jiang Zhang 1, Bo Li 1,* and Shengbo Chen 2
1Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of
Space Sciences, Shandong University, Weihai 264200, China; ythuang1999@mail.sdu.edu.cn (Y.H.);
zhang_jiang@sdu.edu.cn (J.Z.)
2College of Geoexploration Science and Technology, Jilin University, Changchun 130000, China;
chensb@jlu.edu.cn
*Correspondence: libralibo@sdu.edu.cn
Abstract: Water ice has been found in the permanently shadowed regions of impact craters around
the lunar South Pole, which makes them ideal areas for in situ exploration missions. However, near
the rim of impact craters, construction and exploration activities may cause slope instability. As a
result, a better understanding of the shear strength of lunar soil under higher stress conditions is
required. This paper mainly uses the finite element method to analyze slope stability to determine
the position and shape of the slip surface and assess the safety factor. The height and gradient of
the slope, the shear strength of lunar soil, and the lunar surface mission all influence the stability
of the slope. We also analyze the soil mechanical properties of a soil slope adjacent to the traverse
path of the Chang’E-4 Yutu-2 rover. Determining the stability of the slope at the lunar South Pole
impact crater under various loading conditions will enhance the implementation of the lunar surface
construction program. In this respect, this paper simulates a lunar mission landing at the Shackleton
and Shoemaker craters and indicates that areas with higher cohesion lunar soil may be more stable
for exploration in the more complex terrain of the South Pole.
Keywords: shear strength; slope stability; impact crater simulant; lunar South Pole; Chang’E-4
1. Introduction
Radar measurements from the Clementine mission have revealed the possibility of
water ice in a permanently shadowed region of the lunar South Pole [
1
]. Neutron spec-
trometers on the Lunar Reconnaissance Orbiter (LRO) detected abundant hydrogen in
permanently shadowed regions (PSRs), which were also able to infer the presence of water
ice [
2
]. The discovery of water ice in PSRs is of great significance for understanding the
formation and evolution of the Moon and, even, the Earth. Therefore, more and more
national space agencies plan to launch a series of missions to detect near-surface volatiles
at the lunar South Pole in the future. For example, China’s Chang’E-7 mission is expected
to land at the lunar South Pole to conduct polar environmental and resource surveys, and
Russia’s Luna-25 and Luna-26 missions also target this region [
3
]. Knowledge of the lunar
South Pole environment and the physical mechanics is essential for subsequent landing
and sampling missions. A feature of the lunar polar regions is complex terrain and many
permanently shadowed impact craters that may contain water ice. For example, the polar
region has dozens of distributed impact craters, most of which have a diameter of more
than 10 km and a depth of about 3 km, as shown in Figure 1.
The lunar polar region has another unique environmental condition: persistent solar
illumination for more than 100 days [
4
]. Persistent solar illumination is required to provide
a continuous solar energy supply, maintain a suitable thermal environment, and carry
out complex exploration activities for long-term in situ missions targeting impact craters
around the lunar South Pole. As determined in previous studies, high crater rims close
Remote Sens. 2024,16, 371. https://doi.org/10.3390/rs16020371 https://www.mdpi.com/journal/remotesensing

Remote Sens. 2024,16, 371 2 of 17
to the poles generally receive better illumination than other regions. Therefore, in order
to complete the landing of the scientific detection mission and the water ice detection
mission, the landing area should be selected as a high elevation from the continuous light
area, and, under this premise, should be as close as possible to the PSRs to facilitate the
implementation of the water ice detection mission. In addition, the crater rim is a good
choice, which not only satisfies the above conditions, but is also relatively flat and higher
than the surrounding terrain to avoid blocking the sun. For example, the Russian lunar
Earth mission and the Chinese Chang’E-7 mission have chosen the crater rim as a candidate
landing area [
5
]. However, the crater wall adjacent to the rims is steeply sloped and covered
by fine, loose soils, which frequently results in landslides and slope collapses [
6
–
8
], posing
a great challenge to exploration activities. Therefore, it is of great significance to study the
physical and mechanical properties of the lunar regolith at the South Pole and the stability
of the impact crater slope to evaluate the safety of the pre-selected landing zone. In this
paper, we will evaluate the safety of a landing mission by simulating loadings on the rims
of polar impact crater slopes using the finite element method, so as to provide a reference
on the physical properties of the lunar regolith for future projects.
Remote Sens. 2024, 16, x FOR PEER REVIEW 2 of 19
craters around the lunar South Pole. As determined in previous studies, high crater rims
close to the poles generally receive beer illumination than other regions. Therefore, in
order to complete the landing of the scientific detection mission and the water ice detec-
tion mission, the landing area should be selected as a high elevation from the continuous
light area, and, under this premise, should be as close as possible to the PSRs to facilitate
the implementation of the water ice detection mission. In addition, the crater rim is a good
choice, which not only satisfies the above conditions, but is also relatively flat and higher
than the surrounding terrain to avoid blocking the sun. For example, the Russian lunar
Earth mission and the Chinese Chang’E-7 mission have chosen the crater rim as a candi-
date landing area [5]. However, the crater wall adjacent to the rims is steeply sloped and
covered by fine, loose soils, which frequently results in landslides and slope collapses [6–
8], posing a great challenge to exploration activities. Therefore, it is of great significance
to study the physical and mechanical properties of the lunar regolith at the South Pole and
the stability of the impact crater slope to evaluate the safety of the pre-selected landing
zone. In this paper, we will evaluate the safety of a landing mission by simulating loadings
on the rims of polar impact crater slopes using the finite element method, so as to provide
a reference on the physical properties of the lunar regolith for future projects.
Figure 1. (a) The map from LROC NAC; high-resolution images (0.5 m/pixel) of the lunar South
Pole (85°S to South Pole). (b) The LOLA South Pole DEM (20 m/pixel).
In this paper, based on high-resolution Digital Elevation Model (DEM) data and nar-
row-angle camera (NAC) images from the Lunar Reconnaissance Orbiter Camera
(LROC)[9], we used the finite element method (FEM) to study the slope stability of the
crater slope in the lunar South Pole. In Section 2, the principle of the finite element strength
reduction method is provided and we introduce the selection of the model parameters. In
Section 3, the relationships between the angles, height of the crater slope, shear strength,
and surface loadings with slope stability are discussed. In Section 4, an impact crater near
the actual patrol path of the Yutu-2 rover is selected to establish a two-dimensional slope
stability model. Then, the physical characteristics of lunar soil in the Chang’E-4 landing
area are studied. In the last section, based on the high-resolution DEM data, the two-di-
mensional slope stability models are established at the Shoemaker and Shackleton impact
Figure 1. (a) The map from LROC NAC; high-resolution images (0.5 m/pixel) of the lunar South Pole
(85◦S to South Pole). (b) The LOLA South Pole DEM (20 m/pixel).
In this paper, based on high-resolution Digital Elevation Model (DEM) data and
narrow-angle camera (NAC) images from the Lunar Reconnaissance Orbiter Camera
(LROC) [
9
], we used the finite element method (FEM) to study the slope stability of the
crater slope in the lunar South Pole. In Section 2, the principle of the finite element strength
reduction method is provided and we introduce the selection of the model parameters. In
Section 3, the relationships between the angles, height of the crater slope, shear strength,
and surface loadings with slope stability are discussed. In Section 4, an impact crater
near the actual patrol path of the Yutu-2 rover is selected to establish a two-dimensional
slope stability model. Then, the physical characteristics of lunar soil in the Chang’E-4
landing area are studied. In the last section, based on the high-resolution DEM data, the
two-dimensional slope stability models are established at the Shoemaker and Shackleton
impact crater slopes to evaluate the influence of the weight of the scientific research station
on the stability of the impact crater slope.

Remote Sens. 2024,16, 371 3 of 17
2. Materials and Methods
The surface and body force system applied to the internal slope of the impact crater
can cause changes in the stresses and strains inside the crater, even to the limit of its shear
strength, resulting in irreversible plastic deformation. Failure of a slope frequently takes
the form of quite a large sliding of the deformation part, resulting in a lower boundary
deep within the soil mass, called the failure surface. The study of slope stability mainly
focuses on the failure surface of the slope and the corresponding factor of slope safety.
2.1. Data
In June 2009, the National Aeronautics and Space Administration (NASA) launched
the Lunar Reconnaissance Orbiter (LRO), which is still orbiting the Moon at an altitude
of 50–200 km. The LROC is a three-camera system mounted on the LRO that can be used
to capture high-resolution black-and-white images and medium-resolution multispectral
images of the lunar surface [
9
]. In Figure 1a, most of the map is drawn using high-resolution
images, and the blanks are supplemented by medium-resolution images. The NAC images
around the Chang’E-4 landing area were used for obtaining information on the impact
crater slope that can be downloaded from the website https://wms.lroc.asu.edu/lroc/
view_lroc/LRO-L-LROC-3-CDR-V1.0/M1303619844LC (accessed on 28 December 2021).
The Yutu-2 rover carries a panoramic stereo camera (Pancam) to acquire three-dimensional
terrain information, which can deliver a high-resolution DEM (0.5 m/pixel) of the Chang’E-4
landing site [
10
]. The DEM (0.5 m/pixel) from the Pancam was used for extracting the height
and angle of a soil slope adjacent to the Yutu-2 rover traverse path, as detailed in Section 4.
LOLA is a lunar orbital laser altimeter that provides high-resolution DEM images to determine
the global topography of the lunar surface. The LOLA DEM of 20 m/pixel, as shown in
Figure 1b, was used to describe the DEM map from 85
◦
S to 90
◦
S. The LOLA DEM at the 5 m
pixel scale, as the highest spatial resolution, was used for extracting the slope angles of the
Shoemaker and Shackleton craters.
2.2. Finite Element Strength Reduction Method
Slope stability analysis has always been a particularly complicated and important issue
in geotechnical engineering on Earth. The traditional method of slope stability analysis
needs to assume a reasonable failure surface, divide the rock and soil mass on the failure
surface into multiple soil strips, establish a balance equation on the force and moment
between the soil strips, and obtain the factor of safety (FoS) corresponding to the failure
surface [
11
]. It is necessary to assume multiple failure surfaces and then find a particular
surface for which the FoS is the smallest [
12
]. Recently, the FEM has also helped solve slope
stability analysis; the concept of the FoS has a certain significance in the strength of the
reserves. According to the Mohr–Coulomb criterion and Bishop’s definition [
11
–
14
], the
FoS can be expressed as:
FoS =R1
0(c+tan φ)dl
R1
0τdl (1)
Divide both sides of Equation (1) by the FoS:
FoS =R1
0c
FoS +σtan φ
FoS dl
R1
0τdl =R1
0(c′+σtan φ′)dl
R1
0τdl (2)
where
c′=c
FoS
,
tan φ′=tan φ
FoS
. Where
σ
is positive stress, and
τ
is the shear strength. Where
cand
φ
are the actual shear strength parameters of the soil body, which are the cohesion
and internal friction angle, respectively. In addition to this, c
′
and
φ′
are the shear strength
parameters of the lunar soil body when it reaches the limit state and are the cohesion and
internal friction angle after reduction.

Remote Sens. 2024,16, 371 4 of 17
As early as 1975, Zienkiewice et al. proposed the strength reduction method (SRM) for
slope stability analysis [
13
]. Later, it was used for applying different FoS to the c
′
and tan
φ′
terms [
15
]. According to the SRM procedure, c
′
and tan
φ′
are progressively reduced
using the strength reduction factor (SRF) until the ultimate damage state is reached. When
reaching the limit state, the node of the model displacement increases sharply. At this time,
the corresponding SRF is the minimum FoS in the whole slope, as shown by Equation (3):
FoS =c
c′=tan φ
tan φ′(3)
When the FoS equals 1, the slope is in a critical equilibrium state. When the FoS is
greater than 1, the slope is damaged. When the FoS is less than 1, the slope does not change
compared with the initial state.
2.3. Two-Dimensional Model of Slope Stability of Lunar Impact Crater
On Earth, the FEM is very commonly and widely used to study the stability of a soil
slope [
16
] and has been verified using experiments [
17
]. This method can be applied to
the complex geological activity of soil composition and the terrain environment [
17
–
21
].
With the development of the network combined with deep learning, it predicts engineering
safety and disaster impacts [
22
]. In this study, finite element software was applied to
establish the stability model of the impact crater slope at the lunar South Pole. The first
step is to construct a geometric model that takes into account the shape of the crater. The
topography of the Moon is predominantly characterized by impact craters, which exhibit
variations in terms of size, depth, and slope. Additionally, the crater is coated with a layer
of loose, fine-grained regolith. Bart et al. (2011) found that the depth of the lunar regolith
is typically 2–8 m [
23
]. Li et al. (2020) unveiled that the ejecta deposit covering the von
Kármán impact crater can reach 12 m, using data from the Chang’E-4 Lunar Penetrating
Radar [
24
]. Thus, the determination of the slope height for the model requires consideration
of the thickness of the lunar soil on the slope of the impact crater.
The physical parameters and mechanical properties of lunar soil differ significantly
from those of Earth soil. When the finite element analysis method is used to solve the
slope stability problem on the Moon, the accuracy and rationality of the physical and
mechanical parameters of the lunar soil should be considered to reduce the error caused
by the SRM. The gravity on the lunar surface is only 1/6 that of the Earth. When using
finite element software, it is necessary to set not only the gravity condition of the whole
model, but also the bulk density of the lunar soil. According to previous studies, the
bulk density of lunar soil varies with the depth and position, and the range is about
1.3–2.4 g/cm
3
[
25
]. The bulk density of lunar soil at a depth of 5.8 m was 1.9 g/cm
3
and
1.67 g/cm
3
, respectively, in the patrol path measured by the Yutu-2 radar [
26
]. Therefore,
the lunar soil bulk density is selected as 1.9 g/cm
3
in the stability analysis of the lunar
soil slope in our models. In addition, the stiffness parameters of soil, including Young’s
modulus and Poisson’s ratio, are also very important to analyze in the model. Young’s
modulus and Poisson’s ratio, obtained using the triaxial test, range from 10.3 to 80.8 MPa
and 0.43 to 0.47, respectively [
25
]. Therefore, when establishing the model, the values of
Young’s modulus and Poisson’s ratio are selected as 10 MPa and 0.47.
The shear resistance of lunar soil must be considered in the study of slope instability.
The shear resistance of lunar soil particles is determined by the internal friction angle (
φ
)
and cohesion (c). According to extensive field tests conducted during the Apollo missions
by the United States and the Luna program, as well as tests on actual lunar soil samples
brought back to Earth, the internal friction angle (
φ
) in real lunar soil is 25–50
◦
, and the
cohesion cis 0.26–1.80 kPa [
25
]. The cohesion of simulated basalt in the Chang’E-5 mission,
which simulates lunar soil on Earth, ranges from 2.08 to 5.5 kPa [
27
]. The influence of the
internal friction angle and cohesion on slope stability are discussed in the following section.
A simple two-dimensional lunar impact crater slope model was established to show
the process for determining the slope stability of the impact crater, as shown in Figure 2.

Remote Sens. 2024,16, 371 5 of 17
For example, 2D slope model 1, shown in Figure 3a, is a homogeneous soil slope in a
hypothetical lunar impact crater. The thickness (H) of the lunar soil on the set slope is 10 m,
and the slope angle of the crater (
β
) is 26.6
◦
. Based on the SRM, model 1 calculated the
slope stability in which Poisson’s ratio is 0.47, the elastic modulus is 10 MPa, the cohesion
(c) is 6 kPa, and the internal friction angle (φ) is 30◦. When FV1 is 3.2, the displacement of
the turning point, shown by the arrow in Figure 3c, takes a dramatic turn that indicates the
slope has suffered from instability, as shown in Figure 3b. At this time, the output result
(FoS) of the model is 3.2, which indicates the initial state of the slope is stable. Figure 3c
shows the slope landslide simulation result. It can be clearly seen in Figure 3c that the failure
surface penetrates from the foot of the slope to the top of the slope. The green–yellow–red
areas in Figure 3d represent the location of the failure surface during the critical state of
the slope. The red area represents the location of the most drastic change in displacement
within the slope. The safety of the impact crater slope can then be determined based on
these results. According to equation 3, the cohesion c
′
is 1.875 and the internal friction
angle
φ′
is 10.2
◦
when the slope instability occurs. Therefore, when the shear strength of
the real lunar regolith is higher than that of c′and φ′, the slope remains stable.
Remote Sens. 2024, 16, x FOR PEER REVIEW 5 of 19
by the United States and the Luna program, as well as tests on actual lunar soil samples
brought back to Earth, the internal friction angle (φ) in real lunar soil is 25°–50°, and the
cohesion c is 0.26–1.80 kPa [25]. The cohesion of simulated basalt in the Chang’E-5 mission,
which simulates lunar soil on Earth, ranges from 2.08 to 5.5 kPa [27]. The influence of the
internal friction angle and cohesion on slope stability are discussed in the following sec-
tion.
A simple two-dimensional lunar impact crater slope model was established to show
the process for determining the slope stability of the impact crater, as shown in Figure 2.
For example, 2D slope model 1, shown in Figure 3a, is a homogeneous soil slope in a
hypothetical lunar impact crater. The thickness (H) of the lunar soil on the set slope is 10
m, and the slope angle of the crater (β) is 26.6°. Based on the SRM, model 1 calculated the
slope stability in which Poisson’s ratio is 0.47, the elastic modulus is 10 MPa, the cohesion
(c) is 6 kPa, and the internal friction angle (φ) is 30°. When FV1 is 3.2, the displacement of
the turning point, shown by the arrow in Figure 3c, takes a dramatic turn that indicates
the slope has suffered from instability, as shown in Figure 3b. At this time, the output
result (FoS) of the model is 3.2, which indicates the initial state of the slope is stable. Figure
3c shows the slope landslide simulation result. It can be clearly seen in Figure 3c that the
failure surface penetrates from the foot of the slope to the top of the slope. The green–
yellow–red areas in Figure 3d represent the location of the failure surface during the crit-
ical state of the slope. The red area represents the location of the most drastic change in
displacement within the slope. The safety of the impact crater slope can then be deter-
mined based on these results. According to equation 3, the cohesion c’ is 1.875 and the
internal friction angle φ’ is 10. 2° when the slope instability occurs. Therefore, when the
shear strength of the real lunar regolith is higher than that of c’ and φ’, the slope remains
stable.
Figure 2. Process for determining the slope stability of the impact crater in the lunar South Pole.
Figure 2. Process for determining the slope stability of the impact crater in the lunar South Pole.
Remote Sens. 2024, 16, x FOR PEER REVIEW 6 of 19
Figure 3. (a) Two-dimensional geometric slope model 1. H is the slope height and β is the slope
angle. (b) A schematic diagram of the relationship between characteristic point displacement and
reduction coefficient. FV1 is the SRF of the model, and displacement is the deformation of the turn-
ing point. (c) In a schematic diagram of slope deformation, PEMGA is the value of the deformation
of the turning point; the red area represents the location of the most drastic change in displacement
within the slope. (d) A cloud diagram of equivalent plastic strain.
3. Stability Analysis on Soil Slope of Lunar Impact Crater
Instability in the soil slope of lunar impact craters can occur in two situations: (1)
shear strength reduction in the impact crater’s soil slope without load, such as the lunar
rover; (2) stress balance failure of the impact crater’s soil slope influenced by load pressure
and distance from the load to the slope top. In this section, we assess the FoS and the
stability of unloaded/loaded soil slopes in impact craters under different parameters.
3.1. Stability Analysis of Unloaded Soil Slopes
The stability of an unloaded soil slope of an impact crater depends on its shear
strength, which is affected by many factors, such as the slope angle, slope height, cohesion,
and internal friction angle. In this paper, a two-dimensional geometric model was estab-
lished of an unloaded soil slope and used to assess how these factors affect the FoS and
slope stability.
3.1.1. Slope Stability Affected by Different Slope Angles
This section studies the stability of unloaded soil slopes at different slope angles,
while keeping the slope height, cohesion, and the internal friction angle at certain values.
Firstly, a two-dimensional geometric model is established of an unloaded soil slope, in
which the slope height (H) is 10 m, the cohesion (c) is 6 kPa, and the internal friction angle
(φ) is 30°. Then, the FoS of the slopes are calculated at slope angles of 63.4°, 45°, 31°, and
26.6°, respectively (Table 1). The results show that the FoS is lower when increasing the
slope angle, indicating that the slope tends toward instability.
Figure 3. (a) Two-dimensional geometric slope model 1. His the slope height and
β
is the slope angle.
(b) A schematic diagram of the relationship between characteristic point displacement and reduction
coefficient. FV1 is the SRF of the model, and displacement is the deformation of the turning point. (c) In a

Remote Sens. 2024,16, 371 6 of 17
schematic diagram of slope deformation, PEMGA is the value of the deformation of the turning point;
the red area represents the location of the most drastic change in displacement within the slope. (d) A
cloud diagram of equivalent plastic strain.
3. Stability Analysis on Soil Slope of Lunar Impact Crater
Instability in the soil slope of lunar impact craters can occur in two situations: (1) shear
strength reduction in the impact crater’s soil slope without load, such as the lunar rover;
(2) stress balance failure of the impact crater’s soil slope influenced by load pressure and
distance from the load to the slope top. In this section, we assess the FoS and the stability
of unloaded/loaded soil slopes in impact craters under different parameters.
3.1. Stability Analysis of Unloaded Soil Slopes
The stability of an unloaded soil slope of an impact crater depends on its shear strength,
which is affected by many factors, such as the slope angle, slope height, cohesion, and
internal friction angle. In this paper, a two-dimensional geometric model was established of
an unloaded soil slope and used to assess how these factors affect the FoS and slope stability.
3.1.1. Slope Stability Affected by Different Slope Angles
This section studies the stability of unloaded soil slopes at different slope angles, while
keeping the slope height, cohesion, and the internal friction angle at certain values. Firstly,
a two-dimensional geometric model is established of an unloaded soil slope, in which the
slope height (H) is 10 m, the cohesion (c) is 6 kPa, and the internal friction angle (
φ
) is
30
◦
. Then, the FoS of the slopes are calculated at slope angles of 63.4
◦
, 45
◦
, 31
◦
, and 26.6
◦
,
respectively (Table 1). The results show that the FoS is lower when increasing the slope
angle, indicating that the slope tends toward instability.
Table 1. FoS for different slope angles, slope heights, levels of cohesion, and friction angles.
Parameter (Unit) Value FoS
Slope angle (◦)
63.4 1.78
45.0 2.27
33.7 2.85
26.6 3.11
Slope height (m)
12 2.02
10 2.27
8 2.65
6 3.14
Cohesion (kPa)
6 2.27
9 2.93
12 3.55
Internal friction angle (◦)
15 1.71
20 1.93
25 2.10
30 2.27
The failure surfaces of unloaded soil slopes at different slope angles are shown in
Figure 4. We can see that the green–yellow–red areas, representing the location of the
failure surface, developed from the toe to the shoulder of the slope. The failure surfaces
move deeper and away from the toe of the slopes, and their areas become larger with a
decreasing slope angle.

Remote Sens. 2024,16, 371 7 of 17
Remote Sens. 2024, 16, x FOR PEER REVIEW 8 of 19
Figure 4. Failure surfaces (shown in green–yellow–red) and mesh deformation of unloaded soil
slopes at slope angles of 63.4° (a), 45° (b), 31° (c), and 26.6° (d).
3.1.2. Slope Stability Affected by Different Slope Heights
This section studies the stability of unloaded soil slopes with different slope heights,
while keeping the slope angle, cohesion, and internal friction angle at certain values. A
two-dimensional geometric model is established of an unloaded soil slope, in which the
slope angle (β) is 45°, the cohesion (c) is 6 kPa, and the internal friction angle (φ) is 30°.
The lunar regolith thickness has been studied by drilling, microwave remote sensing, and
topography and spectral statistics of small crater ejecta, revealing a range from 2 m in
young maria to more than 10 m in old highlands [26,28–30]. Therefore, the slope heights
selected in this section are 12 m, 10 m, 8 m, and 6 m, for which the FoS are calculated,
respectively (Table 1). The results show that the FoS is lower when increasing the slope
height, indicating that the slope is more prone to instability. The failure surfaces of un-
loaded soil slopes with different slope heights are shown in Figure 5. We can see that the
failure surfaces move deeper and away from the toe of the slopes with decreasing soil
thickness.
Figure 4. Failure surfaces (shown in green–yellow–red) and mesh deformation of unloaded soil
slopes at slope angles of 63.4◦(a), 45◦(b), 31◦(c), and 26.6◦(d).
3.1.2. Slope Stability Affected by Different Slope Heights
This section studies the stability of unloaded soil slopes with different slope heights,
while keeping the slope angle, cohesion, and internal friction angle at certain values. A
two-dimensional geometric model is established of an unloaded soil slope, in which the
slope angle (
β
) is 45
◦
, the cohesion (c) is 6 kPa, and the internal friction angle (
φ
) is 30
◦
.
The lunar regolith thickness has been studied by drilling, microwave remote sensing, and
topography and spectral statistics of small crater ejecta, revealing a range from 2 m in young
maria to more than 10 m in old highlands [
26
,
28
–
30
]. Therefore, the slope heights selected
in this section are 12 m, 10 m, 8 m, and 6 m, for which the FoS are calculated, respectively
(Table 1). The results show that the FoS is lower when increasing the slope height, indicating
that the slope is more prone to instability. The failure surfaces of unloaded soil slopes with
different slope heights are shown in Figure 5. We can see that the failure surfaces move
deeper and away from the toe of the slopes with decreasing soil thickness.
3.1.3. Slope Stability Affected by Different Soil Cohesion and Internal Friction Angles
This section will study the stability of unloaded soil slopes with different cohesion
and friction angles, while keeping the slope angle and soil thickness at certain values.
A two-dimensional geometric model is established of an unloaded soil slope, in which
the slope angle (
β
) is 45
◦
and the slope height (H) is 10 m. The FoS are calculated at an
internal friction angle of 30
◦
and at cohesion values of 6 kPa, 9 kPa, and 12 kPa, respectively
(Table 1). The FoS is higher when the cohesion is increased, indicating the slope is more
stable. The failure surfaces of unloaded soil slopes with different levels of soil cohesion

Remote Sens. 2024,16, 371 8 of 17
are shown in Figure 6. They move deeper and away from the toe and the shoulder of the
slopes with increasing soil cohesion.
Remote Sens. 2024, 16, x FOR PEER REVIEW 9 of 19
Figure 5. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes, with a soil thickness
of 12 m (a), 10 m (b), 8 m (c), and 6 m (d).
3.1.3. Slope Stability Affected by Different Soil Cohesion and Internal Friction Angles
This section will study the stability of unloaded soil slopes with different cohesion
and friction angles, while keeping the slope angle and soil thickness at certain values. A
two-dimensional geometric model is established of an unloaded soil slope, in which the
slope angle (β) is 45° and the slope height (H) is 10 m. The FoS are calculated at an internal
friction angle of 30° and at cohesion values of 6 kPa, 9 kPa, and 12 kPa, respectively (Table
1). The FoS is higher when the cohesion is increased, indicating the slope is more stable.
The failure surfaces of unloaded soil slopes with different levels of soil cohesion are shown
in Figure 6. They move deeper and away from the toe and the shoulder of the slopes with
increasing soil cohesion.
Figure 6. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes, with a cohesion of
6 kPa (a), 9 kPa (b), and 12 kPa (c).
Figure 5. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes, with a soil thickness
of 12 m (a), 10 m (b),8m(c), and 6 m (d).
Remote Sens. 2024, 16, x FOR PEER REVIEW 9 of 19
Figure 5. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes, with a soil thickness
of 12 m (a), 10 m (b), 8 m (c), and 6 m (d).
3.1.3. Slope Stability Affected by Different Soil Cohesion and Internal Friction Angles
This section will study the stability of unloaded soil slopes with different cohesion
and friction angles, while keeping the slope angle and soil thickness at certain values. A
two-dimensional geometric model is established of an unloaded soil slope, in which the
slope angle (β) is 45° and the slope height (H) is 10 m. The FoS are calculated at an internal
friction angle of 30° and at cohesion values of 6 kPa, 9 kPa, and 12 kPa, respectively (Table
1). The FoS is higher when the cohesion is increased, indicating the slope is more stable.
The failure surfaces of unloaded soil slopes with different levels of soil cohesion are shown
in Figure 6. They move deeper and away from the toe and the shoulder of the slopes with
increasing soil cohesion.
Figure 6. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes, with a cohesion of
6 kPa (a), 9 kPa (b), and 12 kPa (c).
Figure 6. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes, with a cohesion of
6 kPa (a), 9 kPa (b), and 12 kPa (c).
A two-dimensional geometric model is established of an unloaded soil slope, in which
the slope angle (
β
) is 45
◦
and the slope height (H) is 10 m. Then, the FoS is calculated at a
cohesion of 6 kPa and at internal friction angle values of 15
◦
, 20
◦
, 25
◦
, and 30
◦
, respectively
(Table 1). It can be seen in Table 1that the FoS is higher when increasing the internal friction
angle, indicating that the slope is more stable. The failure surfaces of unloaded soil slopes
with different internal friction angles are shown in Figure 7. The failure surfaces move
upwards and towards the slope surface when increasing the soil’s internal friction angle.

Remote Sens. 2024,16, 371 9 of 17
Remote Sens. 2024, 16, x FOR PEER REVIEW 10 of 19
A two-dimensional geometric model is established of an unloaded soil slope, in
which the slope angle (β) is 45° and the slope height (H) is 10 m. Then, the FoS is calculated
at a cohesion of 6 kPa and at internal friction angle values of 15°, 20°, 25°, and 30°, respec-
tively (Table 1). It can be seen in Table 1 that the FoS is higher when increasing the internal
friction angle, indicating that the slope is more stable. The failure surfaces of unloaded
soil slopes with different internal friction angles are shown in Figure 7. The failure sur-
faces move upwards and towards the slope surface when increasing the soil’s internal
friction angle.
Figure 7. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes at internal friction
angles of 15° (a), 20° (b), 25° (c), and 30° (d).
3.2. Stability Analysis of Loaded Soil Slopes
The stability of a loaded soil slope of an impact crater is affected by the load pressure
and loading position. We will also establish a two-dimensional geometric model of a
loaded soil slope and a simplified model of the Chang’E-4 Yutu-2 rover as a load and,
then, evaluate how the load pressure and loading position affect the safety factor and
slope stability.
The Yutu-2 rover has a mass of 135 kg and employs a six-wheel drive, each of which
is 0.3 m in diameter and 0.15 m wide. The three wheels on the left or right side are evenly
separated at a distance of 0.75 m [31]. The effective wheel/regolith contact surface area S
can be estimated by the wheel diameter D, width d, and sinkage z using Equation (4):
−
−
=

12
cos 2
Dz
SdD D (4)
The load pressure exerted on the slope surface by each wheel is:
=6
mg
PS (5)
where m is the rover mass and g (1.63 m/s2) is the acceleration of gravity on the Moon.
Using the wheel sinkage z (0.78 m) from Tang (2020) [31], the load pressure P is about 2.5
Figure 7. Failure surfaces (shown in green–yellow–red) of unloaded soil slopes at internal friction
angles of 15◦(a), 20◦(b), 25◦(c), and 30◦(d).
3.2. Stability Analysis of Loaded Soil Slopes
The stability of a loaded soil slope of an impact crater is affected by the load pressure
and loading position. We will also establish a two-dimensional geometric model of a
loaded soil slope and a simplified model of the Chang’E-4 Yutu-2 rover as a load and,
then, evaluate how the load pressure and loading position affect the safety factor and
slope stability.
The Yutu-2 rover has a mass of 135 kg and employs a six-wheel drive, each of which is
0.3 m in diameter and 0.15 m wide. The three wheels on the left or right side are evenly
separated at a distance of 0.75 m [
31
]. The effective wheel/regolith contact surface area S
can be estimated by the wheel diameter D, width d, and sinkage zusing Equation (4):
S=dD cos−1D/2 −z
D/2 (4)
The load pressure exerted on the slope surface by each wheel is:
P=mg
6S(5)
where mis the rover mass and g(1.63 m/s
2
) is the acceleration of gravity on the Moon.
Using the wheel sinkage z(0.78 m) from Tang (2020) [
31
], the load pressure Pis about
2.5 kPa. In a two-dimensional geometric model of a loaded soil slope, the contact points
between the three wheels on the same side and the lunar regolith are loading positions, as
shown by the arrows in Figure 8a.
3.2.1. Slope Stability Analysis Using Different Loading Pressures
Firstly, the stability of a loaded soil slope of an impact crater with different load
pressures was discussed. A two-dimensional geometric model is established of the loaded
soil slope of an impact crater, in which Poisson’s ratio is 0.47, the elastic modulus is 10 MPa,
the cohesion (c) is 6 kPa, the internal friction angle (
φ
) is 30
◦
, the slope angle (
β
) is 45
◦
, and
the slope height (H) is 10 m. Assuming the front wheels of the Yutu-2 rover stop on the top
of the slope, the FoS is calculated at load pressures of 2.5 kPa, 5 kPa, 10 kPa, and 15 kPa,
respectively (Table 2).

Remote Sens. 2024,16, 371 10 of 17
Remote Sens. 2024, 16, x FOR PEER REVIEW 11 of 19
kPa. In a two-dimensional geometric model of a loaded soil slope, the contact points be-
tween the three wheels on the same side and the lunar regolith are loading positions, as
shown by the arrows in Figure 8a.
Figure 8. Failure surfaces (shown in green–yellow–red) and mesh deformation of loaded soil slopes
under load pressures of 2.5 kPa (a), 5 kPa (b), 10 kPa (c), and 15 kPa (d), the black arrows indicate
the loading model.
3.2.1. Slope Stability Analysis Using Different Loading Pressures
Firstly, the stability of a loaded soil slope of an impact crater with different load pres-
sures was discussed. A two-dimensional geometric model is established of the loaded soil
slope of an impact crater, in which Poisson’s ratio is 0.47, the elastic modulus is 10 MPa,
the cohesion (c) is 6 kPa, the internal friction angle (φ) is 30°, the slope angle (β) is 45°, and
the slope height (H) is 10 m. Assuming the front wheels of the Yutu-2 rover stop on the
top of the slope, the FoS is calculated at load pressures of 2.5 kPa, 5 kPa, 10 kPa, and 15
kPa, respectively (Table 2).
Figure 8. Failure surfaces (shown in green–yellow–red) and mesh deformation of loaded soil slopes
under load pressures of 2.5 kPa (a), 5 kPa (b), 10 kPa (c), and 15 kPa (d), the black arrows indicate the
loading model.
Table 2. FoS for different load pressures and load positions.
Parameter (Unit) Value FoS
P(MPa)
2.5 2.09
5 1.94
10 1.83
15 1.71
L(m)
0 2.04
3 2.14
5 2.54
The FoS is lower when the load pressure is increased, indicating the slope tends toward
instability (Table 2). The failure surfaces of loaded soil slopes under different load pressures
are shown in Figure 8. The maximum strain in the plastic zone shifts from the toe of the
slope toe to the loading position, while increasing the load pressure. Meanwhile, the top of
the slope is damaged more severely, causing subsidence of the slope shoulder.

Remote Sens. 2024,16, 371 11 of 17
3.2.2. Slope Stability Analysis Using Different Loading Positions
In this section, the stability of loaded soil slopes in impact craters for different loading
positions is discussed, which are defined as the distance between the front wheels of the
rover and the top of the slope in an impact crater. A two-dimensional geometric model is
established of a loaded soil slope, in which Poisson’s ratio is 0.47, the elastic modulus is
10 MPa, the cohesion (c) is 6 kPa, the internal friction angle (
φ
) is 30
◦
, the slope angle (
β
) is
45
◦
, the slope height (H) is 10 m, and the load pressure (P) of the Yutu-2 rover is 2.5 kPa.
The FoS is calculated at loading positions of 0 m, 3 m, and 6 m, respectively (Table 2).
As the rover moves away from the top of the slope, the FoS is higher, and the slope
is more stable (Table 2). Meanwhile, the failure point on the shoulder moves away from
the top of the slope, which causes subsidence of the failure surface and an increase in the
landslide volume (Figure 9).
Remote Sens. 2024, 16, x FOR PEER REVIEW 13 of 19
Figure 9. Failure surfaces (shown in green–yellow–red) and mesh deformation of loaded soil slopes,
with loading positions of 0 m (a), 3 m (b), and 6 m (c), the black arrows indicate the loading model.
4. Soil Mechanical Properties Estimation of a Soil Slope Adjacent to the Traverse Path
of the Chang’E-4 Yutu-2 Rover
The Chang’E-4 mission landed in the Von Kármán crater on the far side of the Moon
on 3 January 2019 [32]. Most of the landing area is flat, with a slope less than 15° [28].
Without a dedicated payload on the Yutu-2 rover, it is hard to directly obtain the soil shear
strength of the flat area. However, we can establish a two-dimensional geometric model
of the slope of a real lunar impact crater in the portal area of the Yutu-2 rover and estimate
the lower limits of the cohesion and internal friction angle of the lunar soils.
The Chang’E-4 landing area and the traverse path of the Yutu-2 rover are shown in
Figure 10a. There are fresh craters with sharp rim structures and highly degraded craters
that are nearly beyond recognition. We selected a soil slope with a flat shoulder in a crater
adjacent to the traverse path (Figure 10b), with a slope angle of 18.4° and a height of 2 m.
A two-dimensional geometric model is established of the soil slope, in which Poisson’s
Figure 9. Failure surfaces (shown in green–yellow–red) and mesh deformation of loaded soil slopes,
with loading positions of 0 m (a),3m(b), and 6 m (c), the black arrows indicate the loading model.

Remote Sens. 2024,16, 371 12 of 17
4. Soil Mechanical Properties Estimation of a Soil Slope Adjacent to the Traverse Path of
the Chang’E-4 Yutu-2 Rover
The Chang’E-4 mission landed in the Von Kármán crater on the far side of the Moon
on 3 January 2019 [
32
]. Most of the landing area is flat, with a slope less than 15
◦
[
28
].
Without a dedicated payload on the Yutu-2 rover, it is hard to directly obtain the soil shear
strength of the flat area. However, we can establish a two-dimensional geometric model of
the slope of a real lunar impact crater in the portal area of the Yutu-2 rover and estimate
the lower limits of the cohesion and internal friction angle of the lunar soils.
The Chang’E-4 landing area and the traverse path of the Yutu-2 rover are shown in
Figure 10a. There are fresh craters with sharp rim structures and highly degraded craters
that are nearly beyond recognition. We selected a soil slope with a flat shoulder in a crater
adjacent to the traverse path (Figure 10b), with a slope angle of 18.4
◦
and a height of 2 m. A
two-dimensional geometric model is established of the soil slope, in which Poisson’s ratio
is 0.47, the elastic modulus is 10 MPa, the load pressure Pof the Yutu-2 rover is 2.5 kPa,
and the loading position is 0 m (Figure 10c).
Remote Sens. 2024, 16, x FOR PEER REVIEW 14 of 19
ratio is 0.47, the elastic modulus is 10 MPa, the load pressure P of the Yutu-2 rover is 2.5
kPa, and the loading position is 0 m (Figure 10c).
Figure 10. (a) The Yutu-2 rover’s traverse path map, with a center coordinate of 177°34′E, 45°27′S.
The base map can be downloaded from hps://wms.lroc.asu.edu/lroc/view_lroc/LRO-L-LROC-3-
CDR-V1.0/M1303619844LC (accessed on 28 December 2021). (b) Topographic profile of P1-P1’. (c)
Failure surfaces’ (shown in green–yellow–red) plastic cloud, the black arrows indicate the loading
model and (d) schematic diagram of relationship between feature point displacement and FoS, the
black arrow indicates the turning point of displacement.
Based on the best estimates of the Apollo model [25], the internal friction angles of
lunar soils range from 30° to 50°, and the cohesion of lunar soils is from 0.1 kPa to 1 kPa.
Therefore, the stability of a soil slope with internal friction angles φ of 30°, 40°, and 50°
and a cohesion c of 0.3 kPa, 0.5 kPa, and 1 kPa, will be discussed. As shown in Table 3, the
FoS derived from these parameter values are all greater than 1, indicating that the Yutu-2
rover can pass by the slope safely, like in real situations.
In our model, the closer the initial value is set to the real situation, the closer the
simulated result is to the real situation when the slope is unstable, and the more likely a
relatively small range in the lower limit value of the internal friction angle can be obtained.
Guo et al. showed that the Chang’E-4 landing site has the same bulk chemical composition
in the regolith (i.e., FeO and Th) as the Apollo 15 soil, among the Apollo and Luna mission
regolith samples [33–35]. Tang et al. also suggested that the Chang’E-4 landing site has
greater soil strength and is close to the Apollo 15 landing site in this regard [31]. In the
Apollo 15 missions, self-recording penetrometer (SRP) measurement and simulation stud-
ies indicated that higher cohesion should apply to higher internal friction angles; for ex-
ample, the cohesion would have to be 1 kPa for a friction angle of 50° [25]. It can be seen
in Figure 10d that the FoS is 4.524 at the turning point of feature point displacement. Table
3 shows that at this initial value, when the slope fails, the cohesion is 0.235 kPa, and the
Figure 10. (a) The Yutu-2 rover’s traverse path map, with a center coordinate of 177
◦
34
′
E, 45
◦
27
′
S.
The base map can be downloaded from https://wms.lroc.asu.edu/lroc/view_lroc/LRO-L-LROC-
3-CDR-V1.0/M1303619844LC (accessed on 28 December 2021). (b) Topographic profile of P1-P1’.
(c) Failure surfaces’ (shown in green–yellow–red) plastic cloud, the black arrows indicate the loading
model and (d) schematic diagram of relationship between feature point displacement and FoS, the
black arrow indicates the turning point of displacement.
Based on the best estimates of the Apollo model [
25
], the internal friction angles of
lunar soils range from 30
◦
to 50
◦
, and the cohesion of lunar soils is from 0.1 kPa to 1 kPa.
Therefore, the stability of a soil slope with internal friction angles
φ
of 30
◦
, 40
◦
, and 50
◦
and
a cohesion cof 0.3 kPa, 0.5 kPa, and 1 kPa, will be discussed. As shown in Table 3, the FoS
derived from these parameter values are all greater than 1, indicating that the Yutu-2 rover
can pass by the slope safely, like in real situations.

Remote Sens. 2024,16, 371 13 of 17
Table 3. FoS with the friction angle and cohesion of instability (
φ′
and c
′
) for different friction angles
(c) and levels of cohesion (φ).
φc(kPa) FoS φ′c′(kPa)
30◦
0.5 2.000 16.110◦0.250
0.8 2.465 13.189◦0.325
1 2.679 12.168◦0.373
40◦0.8 3.093 15.186◦0.259
1 3.342 14.101◦0.299
50◦0.8 3.450 19.066◦0.232
1 4.524 15.658◦0.235
In our model, the closer the initial value is set to the real situation, the closer the
simulated result is to the real situation when the slope is unstable, and the more likely a
relatively small range in the lower limit value of the internal friction angle can be obtained.
Guo et al. showed that the Chang’E-4 landing site has the same bulk chemical composition
in the regolith (i.e., FeO and Th) as the Apollo 15 soil, among the Apollo and Luna mission
regolith samples [
33
–
35
]. Tang et al. also suggested that the Chang’E-4 landing site has
greater soil strength and is close to the Apollo 15 landing site in this regard [
31
]. In the
Apollo 15 missions, self-recording penetrometer (SRP) measurement and simulation studies
indicated that higher cohesion should apply to higher internal friction angles; for example,
the cohesion would have to be 1 kPa for a friction angle of 50
◦
[
25
]. It can be seen in
Figure 10d that the FoS is 4.524 at the turning point of feature point displacement. Table 3
shows that at this initial value, when the slope fails, the cohesion is 0.235 kPa, and the angle
of internal friction is 15.658
◦
. Both values, as two lower limits, are far lower than the Apollo
15 soil values. As a highly degraded impact crater, the impact crater has a small slope angle
and height, which may lead to the stability of the slope.
5. Stability Analysis of Loaded Slopes in Potential Landing Regions of the Lunar
South Pole
Recently, several lunar exploration missions have been planned to investigate the
permanently shadowed regions in the southern polar regions, such as NASA’s Artemis
and the Chinese Chang’E-7 missions. The Artemis I mission plans to land around the lunar
South Pole in 2024, for which a total of 13 candidate landing regions have been selected
along the rims and exterior regions of impact craters, namely, the Faustini, Shackleton,
de Gerlache, Kocher, Haworth, Malapert, Nobile, and Amundsen craters [
36
]. Zhang
et al. (2020) proposed 10 candidate landing areas located in four craters (the Shackleton,
Haworth, Shoemaker, and Cabeus craters) [
37
]. As case studies, two slopes located at the
Shackleton and Shoemaker craters were selected to evaluate their stability, the results of
which can be used in future lunar polar exploration missions.
For a lunar mission landing at the Shackleton and Shoemaker craters, continuous solar
illumination is needed to provide an energy supply and to maintain a suitable thermal
environment. Therefore, flat areas on crater rims are preferred candidate landing sites, as
they are higher than the surrounding terrain and are safe for landing [
4
]. The two slopes
selected for stability analysis are adjacent to flat crater rims, as shown in Figure 11. However,
the two craters have a diameter of more than 10 km and a depth of more than 5 km. Due
to the limited research on the structure of lunar regolith at the lunar South Pole, we have
made a reasonable assumption that the slope height of the models is 10 m based on the
previous studies that large impact craters have thicker lunar regolith.
In the flat area on the rim of the Shackleton crater, the slope angle is 45
◦
, as calculated
using the topographic profile in Figure 11c. A two-dimensional geometric model of the
soil slope is established, in which Poisson’s ratio is 0.47, the elastic modulus is 10 MPa, the
cohesion cis 6 kPa, and the internal friction angle
φ
is 30
◦
. The load pressure, increasing
from 1.5 kPa to 9.0 kPa, is applied at three positions, separated by a distance of 1.5 m. The

Remote Sens. 2024,16, 371 14 of 17
resulting FoS lookup table is shown in Table 4. For instance, the slope can hold its stability
when the lunar soil cohesion is greater than 2.673 kPa and the internal friction angle is
greater than 14.430
◦
, as the load pressure is 1.5 kPa. As the load pressure increases, the
factor of safety decreases, while the corresponding critical cohesion and internal friction
angle increase.
Remote Sens. 2024, 16, x FOR PEER REVIEW 16 of 19
Figure 11. (a) The base map is the WAC_morphology_globe_June2013_dd0_100m from LOLA, and
the elevation data are also from LRO NAC DEM. The area is the Shackleton crater, with central
coordinates of 128.5°E and 89.3°S. The base map in (b) has the same data source, and the area is the
Shoemaker crater, with the central coordinates of 45.1°E and 88.5°S. (c) is the topographic profiles
of P2-P2′, and (d) is the topographic profiles of P3-P3′. The red arrows indicate the crater slopes we
selected.
Tab le 4. FoS with the friction angle (φ’) and cohesion (c’) of instability for different loading pressures
at the Shackleton crater.
P (kPa) 1.5 3 4.5 6 7.5 9
FoS 2.245 2.226 2.202 2.162 2.158 2.14
c’ (kPa) 2.673 2.695 2.725 2.775 2.780 2.804
φ’ 14.430° 14.548° 14.699° 14.959° 14.986° 15.106°
In the flat area on the rim of the Shoemaker crater, the slope angle is 26.6°, as calcu-
lated using the topographic profile in Figure 11d. A two-dimensional geometric model of
the soil slope is established, in which Poisson’s ratio is 0.47, the elastic modulus is 10 MPa,
the cohesion c is 2 kPa, and the internal friction angle φ is 30°. The load pressure, increas-
ing from 1.5 kPa to 9.0 kPa, is applied at three positions, separated by a distance of 1.5 m.
The resulting FoS lookup table is shown in Table 5. For instance, the slope can stabilize
when the lunar soil cohesion is greater than 0.996 kPa, and the internal friction angle is
greater than 16.042°, as the load pressure is 1.5 kPa. As the load pressure increases, the
FoS decreases, while the corresponding critical cohesion and internal friction angle in-
crease.
Figure 11. (a) The base map is the WAC_morphology_globe_June2013_dd0_100m from LOLA, and
the elevation data are also from LRO NAC DEM. The area is the Shackleton crater, with central
coordinates of 128.5◦E and 89.3◦S. The base map in (b) has the same data source, and the area is the
Shoemaker crater, with the central coordinates of 45.1
◦
E and 88.5
◦
S. (c) is the topographic profiles
of P2-P2
′
, and (d) is the topographic profiles of P3-P3
′
. The red arrows indicate the crater slopes
we selected.
Table 4. FoS with the friction angle (
φ′
) and cohesion (c
′
) of instability for different loading pressures
at the Shackleton crater.
P(kPa) 1.5 3 4.5 6 7.5 9
FoS 2.245 2.226 2.202 2.162 2.158 2.14
c′(kPa) 2.673 2.695 2.725 2.775 2.780 2.804
φ′14.430◦14.548◦14.699◦14.959◦14.986◦15.106◦
In the flat area on the rim of the Shoemaker crater, the slope angle is 26.6
◦
, as calculated
using the topographic profile in Figure 11d. A two-dimensional geometric model of the
soil slope is established, in which Poisson’s ratio is 0.47, the elastic modulus is 10 MPa, the
cohesion cis 2 kPa, and the internal friction angle
φ
is 30
◦
. The load pressure, increasing
from 1.5 kPa to 9.0 kPa, is applied at three positions, separated by a distance of 1.5 m. The
resulting FoS lookup table is shown in Table 5. For instance, the slope can stabilize when the
lunar soil cohesion is greater than 0.996 kPa, and the internal friction angle is greater than
16.042
◦
, as the load pressure is 1.5 kPa. As the load pressure increases, the FoS decreases,
while the corresponding critical cohesion and internal friction angle increase.

Remote Sens. 2024,16, 371 15 of 17
Table 5. FoS with the friction angle (
φ′
) and cohesion (c
′
) of instability for different loading pressures
at the Shoemaker crater.
P(kPa) 1.5 3 4.5 6 7.5 9
FoS 2.009 1.997 1.978 1.965 1.950 1.935
c′(kPa) 0.996 1.002 1.011 1.018 1.026 1.034
φ′16.042◦16.133◦16.280◦16.382◦16.501◦16.222◦
As can be seen in the two tables, the internal friction angle of the two crater slopes in
the critical state is within 14–16
◦
, while the internal friction angle of the previous exploration
to the Moon is in the range of 25
◦
to 50
◦
[
25
]. Further, the cohesion of the two crater slopes
in the critical state is within 1–3 kPa, while the estimates of the cohesion are 0.1–2 kPa using
data from the Apollo and Lunokhod missions [
25
]. In this case, the cohesion of the lunar
soil at the landing sites must be a higher value to maintain slope stability. Existing studies
on the lunar surface show that the regolith has low cohesion and high friction angles, but
different water ice contents and lunar soil particle compositions at the lunar South Pole
have different effects on the slope stability of the impact crater. In future studies, we will
add more details on the load sequence and lunar soil simulation and conduct a 3D model
simulation, so as to more accurately simulate the impact of exploration missions on the
lunar surface.
6. Conclusions
Future rovers patrolling the rim of the lunar South Pole to detect water ice could face
more difficulties than the Chang’E-4 mission. In this study, we evaluate landing safety in
the candidate landing area based on remote sensing data, using the finite element method.
We conclude the following:
(1) With the increase in slope angle and slope height, the slope tends to be unstable. With
the increase in cohesion and the internal friction angle of lunar soil, the slope becomes
more stable. The analysis results of the soil slope model under load conditions show
that the soil slope tends to be unstable with the increase in load pressure. When the
rover is in different positions, the further the rover is from the top of the slope, the
more stable the slope.
(2)
The model result based on the true soil slope adjacent to the traverse path shows
that the rover can pass by the slope safely, which is consistent with the facts. This
provides a basis for the application of the FEM on the Moon. The calculated cohesion
and internal friction angles, as two lower limits, are far lower than the Apollo 15 soil
values. As a highly degraded impact crater, the impact crater has a small slope angle
and height, which may lead to the stability of the slope.
(3)
Two-dimensional loaded models based on the Shoemaker and Shackleton craters
were established and FoS lookup tables were created. Based on the FoS lookup tables
for the Shoemaker and Shackleton crater slopes, the ultimate shear strength of the
soil under different load pressures can be found. These will help to better assess the
safety of future missions to explore the Moon. In addition, a higher level of cohesion
of the lunar soil at the landing site is more conducive to maintaining the stability of
the slope.
Author Contributions: Conceptualization, B.L. and J.Z.; methodology, B.L. and Y.H.; software, Y.H.,
S.C. and J.Z.; validation, S.C.; resources, S.C.; writing—original draft preparation, Y.H. and J.Z.;
writing—review and editing, B.L. and S.C.; project administration, B.L.; funding acquisition, S.C. All
authors have read and agreed to the published version of the manuscript.

Remote Sens. 2024,16, 371 16 of 17
Funding: This work is supported by the National Key Research and Development Project
(2019YFE0123300, 2022YFF0711400, 2022YFF0503100), the National Natural Science Foundation
(42372277, 41772346), the Shandong Provincial Natural Science Foundation (ZR2023MD010), and
the Strategic Leading Science and Technology Special Project of the Chinese Academy of Sciences
(XDB41000000).
Data Availability Statement: The NAC and WAC images used in this paper are available at https://wms.
lroc.asu.edu/lroc (accessed on 28 December 2021).
Acknowledgments: The efforts by the science and engineering teams behind all the datasets used in
this study, particularly the LRO mission and NAC and WAC instruments, are gratefully acknowledged.
Conflicts of Interest: The authors declare no conflicts of interest.
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