© The Authors 2024

1 Introduction

A phenomenon called a horizon glow engulfing the Moon was first photographed by Surveyor 5; it was attributed to the scattering of sunlight by charged lunar dust (Rennilson & Criswell 1974). Since then, several probes (e.g., Aollo 15 and 17) indirectly detected the possible dust in the vicinity above the lunar surface (O’Brien 2011). In 1990, the Munich Dust Counter (MDC) on board the HITEN spacecraft recorded approximately 150 dust impacts on the lunar orbit with perilune between 100 and 8000 km and apolune of about 50 000 km, but failed to detect particles near the lunar surface, due to its high orbit (Iglseder et al. 1996). Later, the Lyman-Alpha Mapping Project (LAMP) far-ultraviolet spectrograph carried by the Lunar Reconnaissance Orbiter (LRO) searched for the reported horizon glow, but unfortunately did not observe any dust scattering phenomenon (Feldman et al. 2014). Until 2013, the Lunar Dust EXperiment (LDEX) on board the Lunar Atmosphere and Dust Environment Explorer (LADEE) detected an asymmetric and permanent dust cloud around the Moon (Horányi et al. 2015).

Based on the LDEX measurements, many efforts have been devoted to studying the nature of the lunar dust cloud. Horányi et al. (2015) obtained the size distribution of dust particles, and found the variation in dust density with their altitude and also the local time. Szalay et al. (2019) focused on the impact-ejecta model of particles in the lunar polar regions and derived the dust production rate considering six previously identified meteoroid populations. Later the model was improved by Pokornỳ et al. (2019) via introducing the nightside and dayside imbalance of lunar dust production. Previous studies on the lunar dust dynamics mainly focused on the source of impactors, the production model, and the electrostatic migration of bound particles (e.g., Jones & Brown 1993; Pokornỳ et al. 2019; Hartzell & Scheeres 2013). However, less attention has been paid to particles that can escape the Moon (hereafter referred as escapers). Although most of the dust particles ejected from the lunar surface via hypervelocity impacts eventually fall back (Szalay & Horányi 2016), a small proportion (~2%) of them can reach the farther Earth–Moon space, and can form a steady-state dust torus with a peak number density on the order of 10⁻⁹ m⁻³ (Yang et al. 2022), which is comparable to that of interplanetary dust at 1 AU (~10⁻¹⁰ m⁻³ Grün et al. 1985; Dikarev et al. 2005). Particles may cause a negative effect on astronomical observations for Earth-based observatories (Fladeland 2022).

Several studies have been devoted to analyzing the dynamical transfer of lunar ejecta. Gault (1983) found that 0.5% of lunarejected particles ultimately impact the Earth, and the accretion rate of lunar ejecta onto the Earth is about 10⁷−10⁸ g/yr, corresponding to 10⁻⁴−10⁻³ kg/s. Alexander et al. (1984) analyzed the transport of lunar ejecta in the Earth–Moon system, and inferred that the lunar ejecta flux at the Earth’s magnetopause depends on the lunar phase angle. Gladman et al. (1995) found that 23–50% of lunar-ejected particles are re-accreted by the Earth within 10 Myr. Yamamoto & Mukai (1996) estimated the average speed of particles (~10 km/s) when they reach the near-Earth region, and concluded that the impact angles of particles are nearly isotropic. Fritz (2012) calculated the total mass of lunar ejecta transferred to the Earth in 10⁶ years, which is about 2.0 × 10⁻⁴ kg/s. Fladeland (2022) projected the lunar-ejected particles to the sky of an Earth-based observatory located in the northern hemisphere, and suggested that particles are visible almost every night, with distinct orientations and configurations.

The aim of this work is to investigate the particles ejected from the lunar surface that finally impact the Earth (hereafter referred as Earth impactors), to analyze their orbital characteristics, and to estimate their effects on the Earth-based observation. The paper is organized as follows. In Sect. 2 the dust dynamical model and the simulation scheme shown in our previous paper are presented briefly. In addition, several coordinate systems used in our simulation are introduced. In Sect. 3 the simulation results for particles impacting the Earth are presented, including the fraction of impactors with different initial parameters among all impactors, the distribution of orbital elements at the reaching moment, and the orbital evolution for an Earth impactor. In Sect. 4 lunar-ejected particles are projected to several noted observatories located at different regions of the Earth, to analyze their effect on the Earth-based astronomical observation.

2 Method

2.1 Dust dynamical model and simulation scheme

In our previous work we established the dynamical model and performed the long-term orbital simulations for particles ejected from the lunar surface. Here we briefly introduce the dynamical model and the simulation scheme.

The initial parameters of particles ejected from the lunar surface for the simulations are presented in Table 1. Both the size and the initial velocity of particles follow the power-law distribution, with exponents of 3.7 and 2.2, respectively. To eliminate the seasonal effect, 12 starting moments for particles were chosen, covering one period of the lunar ascending node precession (18.6 Earth years). In addition, particles were assumed to be ejected vertically at 200 starting regions of the lunar surface, corresponding to 10 ejection latitudes and 20 ejection longitudes. The gravity of Earth, solar radiation pressure, Poynting-Robertson drag, and gravity perturbations of the Sun and the Moon were considered in the dynamical model. In total, the motions of 216 000 particles ejected from the lunar surface were simulated until they hit the Moon, hit the Earth, or escaped from the Earth-Moon system, with the maximum motion time (100 Earth years). For more details about the dynamical modeling and the simulation scheme, we refer to Yang et al. (2022).

Fig. 1 Fraction of impactors vs. grain size.

Fig. 2 Fraction of impactors with different launch angles and ejection velocities.

2.2 Coordinate systems used in our work

The integration of the particles’ motion is performed in the J2000 Earth-centered inertial (ECI) frame. Unless otherwise specified, the view angles of particles are provided in the east, north, up (ENU) local tangent plane frame. The coordinates of particles are first transformed from the ECI frame to the Earth-centered, Earth-fixed (ECEF) frame using the sxform_c function from the NAIF SPICE toolkit¹, and then the ECEF coordinates are converted to the ENU coordinates (e.g., Meeus 1991).

3 Particles reaching the Earth

3.1 Effect of initial parameters

By averaging the initial parameters (including the ejection longitude, latitude, velocity, and particle size), the mass rate of Earth impactors was estimated to be 2.3 × 10⁻⁴ kg/s (Yang et al. 2022), which is comparable to the values reported by previous studies: 10⁻⁴−10⁻³ kg/s by Gault (1983) and 2.0 × 10⁻⁴ kg/s by Fritz (2012). For this work we analyzed the simulation results for these lunar-ejected Earth impactors (8077 particles in total).

Figure 1 depicts the fraction of Earth impactors of different sizes. The fraction shows a rapid increase in the size range of [0.2 μm, 2 μm], then drops to approximately 3.6% for particles with the size of 100 μm (Fig. 1). This can be attributed to the size-dependent solar radiation pressure, which can blow out most of the small particles (≤0.5 μm) and leave most of the large particles (≥10 μm) in the dust torus between the Earth and the Moon.

In addition to the grain size, the launch angle (the angle between the ejection velocity of the particle and the motion direction of the Moon; Gladman et al. 1995) and ejection velocity also affect the orbital evolution and the transfer probability of lunar-ejected particles. Figure 2 shows that only particles with ejection velocities in the range [v_esc, 1.27 v_esc] have the chance to impact the Earth, which is reasonable since the low-velocity (<v_esc) particles are unable to escape the lunar gravity, and particles with ejection velocities ≥1.42 v_esc tend to evolve into hyperbolic orbits and leave the Earth-Moon system rapidly. For the launch angle, particles with launch angles >π/2 are likely to transfer from the Moon to the Earth, with the highest transfer probability occurring at the launch angle of π (i.e., when the particles are ejected in the opposite direction of the Moon’s motion around the Earth). Particles with launch angles <π/2 can hardly reach the Earth, which is consistent with Gladman et al. (1995).

Fig. 3 Distribution of a, e, and i for lunar-ejected particles at the moment they reach the Earth. The symbols of a, e, and i denote the semi-major axis, the eccentricity, and the inclination of particles, which are measured in the ECI frame. Panel a: distribution of a. Panel b: distribution of e. Panel c: distribution of i.

Fig. 4 Distribution of travel time of Earth impactors.

3.2 Orbital distribution and evolution

The orbital elements of lunar-ejected particles at the moment of reaching the Earth are shown in Fig. 3. We note that Fig. 3 only includes particles that never leave the Earth’s Hill sphere. The semi-major axes of particles are mainly distributed in the range [20, 150] R_E, with a peak at about 50 R_E. The eccentricities of particles are in the range [0.96, 1], peaking at about 0.99. The inclinations of lunar-ejected particles are dispersed in the range [0°, 180°]. Among them, about 54.0% particles are on prograde orbits, peaking near 40°, while a large number (~46.0%) of particles are on retrograde orbits, peaking near 120°.

The distribution of travel time for lunar-ejected Earth impactors is shown in Fig. 4. Only particles with travel times of less than one year are shown in this figure, which account for ~70.0% of all particles in our simulation. Notably, the travel time in our simulations is longer than those reported by previous studies (e.g., Alexander et al. 1984; Yamamoto & Mukai 1996). It may be attributable to the different grain sizes selected for the simulations between previous studies and this paper (≤0.6 μm in Alexander et al. 1984; <1 μm in Yamamoto & Mukai 1996; 0.2–100 μm in this paper). In our simulations, we note that most of the small impactors (87.2% of 0.2 μm particles and 64.6% of 0.5 μm particles) transfer from the Moon to the Earth within one week, which is consistent with previous studies.

Figure 5 presents the evolution of orbital elements for a 0.2 μm particle from an initial retrograde orbit. The particle is ejected from the lunar surface at a speed of 2.71 km/s; it then decelerates as it climbs out of the Moon’s gravitational well. As it exits the Moon’s Hill sphere, its geocentric eccentricity is 0.97 (see Fig. 5a). Then the particle moves on highly elliptical orbits until it impacts the Earth after about 3.4 days. As Fig. 5b shows, this particle always moves on retrograde orbits during the whole transfer process, and its inclination decreases continuously from the initial 173.5° to 134.2°. The perigee of this particle decreases continuously, while the apogee remains nearly constant around one Earth radius after a rapid descent at the beginning (Fig. 5c).

Fig. 5 Evolution of e, i, q, and Q for a 0.2 μm impactor from initial retrograde orbit, where e, i, q, and Q denote the eccentricity, the inclination, the perigee, and the apogee of the particle, which are measured in the geocentric lunar orbital frame. The dashed lines denote the time at which the particle leaves the Moon’s Hill sphere. Panel a: evolution of e. Panel b: evolution of i. Panel c: evolution of q and Q.

4 Projection onto Earth-based observatories

In order to estimate the effect of lunar-ejected particles on the Earth-based observation, the projection of particles onto several observatories can be obtained by the transformation from the ECI frame to the ENU topocentric frame. Five observatories were selected for this work, the locations of which are listed in Table 2. The lunar-ejected particles are visible (above the local horizon) for all selected observatories, forming dust belts with different orientations and configurations.

Figures 6–10 present the fraction of the solid angle subtended by particles per angular bin from the perspective of different observatories, where the axes are measured in terms of local azimuth and elevation. Figure 6 shows that lunar-ejected particles form a dust belt stretching in the east-west direction, and hardly affect the observation of targets in the northern and southern sky. When observed from the Paranal Observatory of the European Southern Observatory, the lunar-ejected particles form an arc in the northern half of the sky, from about 20° south of east to about due west (see Fig. 7).

To show the influence of the lunar-ejected dust on the polar regions of the Earth, particles are projected into the sky of the IceCube Neutrino Observatory, which is fairly close to the south pole. From this perspective, particles form an arc near the local horizon in the southeastern direction, while the northwestern sky is less dusty (Fig. 8). Since the north pole is primarily covered by the Arctic Ocean, particles are projected to the sky of the Helsinki Observatory (a typical high-latitude observatory located in the northern hemisphere of the Earth), as shown in Fig. 9. Particles are more likely to be observed in the southern sky, forming an arc from due east to about 20° north of west. When observing from the near-equator Quito Astronomical Observatory, observing targets in the southern sky are hardly affected by these lunar-ejected dust particles (Fig. 10).

Figure 11 shows the geometric optical depth and the number density of the lunar-ejected dust particles relative to the Earth-Moon orbital plane, seen from which these particles form a dust torus around the Earth. As Fig. 11b shows, this dust torus is symmetric about the orbital plane, and the number density drops with increasing radial distance from the Earth. We note that the plane of the dust torus does not coincide with the Earth’s equatorial plane, so this torus may exhibit distinct configurations when observed at different observatories and local times.

Additionally, we provide the projection of lunar-ejected particles in the equatorial coordinate system, which is independent of the observation location and time (Fig. 12). Several bright stars are marked in this figure, seen from which stars with low declinations (i.e., Aldebaran, Pollux, Spica, and Antares) are more prone to being obscured by lunar-ejected dust particles.

Fig. 6 Fraction of solid angle subtended by particles per angular bin from the perspective of the Purple Mountain Observatory, at local midnight. The intervals of azimuth and elevation are 2° and 4°, respectively. The outer circle denotes the local horizon.

Fig. 7 Fraction of solid angle subtended by particles per angular bin from the perspective of the Paranal Observatory of the European Southern Observatory, at local midnight.

Fig. 8 Fraction of solid angle subtended by particles per angular bin from the perspective of the IceCube Neutrino Observatory, at local midnight.

Fig. 9 Fraction of solid angle subtended by particles per angular bin from the perspective of the Helsinki Astronomical Observatory, at local midnight.

Fig. 10 Fraction of solid angle subtended by particles per angular bin from the perspective of the Quito Astronomical Observatory, at local midnight.

Fig. 11 Geometric optical depth and number density of lunar-ejected particles relative to the Earth-Moon orbital plane, where the x-axis points to the ascending node of the Moon at the J2000 epoch, the z-axis is along the normal of the orbital plane (north), and the y-axis is determined by the right-handed rule. Panel a: normal geometric optical depth. Panel b: Azimuth-averaged number density in the ρ-z plane, where $\rho =\sqrt{x^2+y^2}$. The blue circle denotes Earth, and the red line denotes the orbit of Moon.

Fig. 12 Fraction of solid angle subtended by particles in the ECI frame. The asterisks denote the bright stars, and the blue asterisks denote the stars that are more prone to being obscured by the lunar-ejected dust particles.

5 Conclusion

In our previous work we adopted the plausible initial distribution and production rate for lunar-ejected particles from the LDEX measurement, and implemented the long-term orbital simulation for these particles. From previous simulation results, we find that about 2.3 × 10⁻⁴ kg/s lunar-ejected particles impact the Earth after long-term orbital evolution, which is a nonnegligible value, and somehow affect the near-Earth space environment.

In this work we provide the simulation results of these Earth impactors. We find that particles ejected from the lunar surface are more likely to reach Earth if they are (1) larger than 1 μm; (2) within the velocity range between 1 v_esc and 1.27 v_esc; (3) ejected in the opposite direction of the Moon’s orbital motion.

We find that most of the lunar-ejected Earth impactors reach the Earth within one year, while most of the small ones (87.2% of 0.2 μm particles and 64.6% of 0.5 μm particles) reach the Earth within one week, which is consistent with previous studies for small impactors (e.g., Alexander et al. 1984; Yamamoto & Mukai 1996). We also find that a small fraction of impactors (about 30.0%) transfer from the Moon to the Earth after a longterm orbital evolution, with a travel time of more than one year. Although these particles only account for a small proportion, they make a great contribution to the dust torus between the Earth and Moon reported by Yang et al. (2022). In addition, a considerable proportion of lunar-ejected Earth impactors can be distinguished from the interplanetary dust particles according to the differences in orbital distributions.

The particles were projected to several Earth-based observatories, and thus the dust distributions from the perspective of different observatories were obtained. For example, particles are most likely to be observed in the northern sky of observatories located in the southern hemisphere of the Earth (e.g., the Paranal Observatory of the European Southern Observatory) and in the opposite direction for the observatories located in the northern hemisphere with high latitude (e.g., the Helsinki Astronomical Observatory). The particles occupy an area in the northern sky of equatorial observatories near the zenith, while they form an arc near the local horizon when observed from the perspective of southern polar regions.

Inevitably, our simulation results are subject to some uncertainties, mainly due to the mass production and the initial distribution of parameters we adopted. In this case, the uncertainties of the fraction of the solid angle subtended by particles and the number density are at least one order of magnitude. However, our results and conclusions are still useful for finding the suitable range of initial parameters for Earth impactors, predicting the visible locations and orientations for lunar-ejected particles, and analyzing their effect on Earth-based observation.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 12311530055, 12002397 and 62388101), and the Shenzhen Science and Technology Program (Grant No. ZDSYS20210623091808026).

References

All Tables

All Figures

	Fig. 1 Fraction of impactors vs. grain size.
In the text

	Fig. 2 Fraction of impactors with different launch angles and ejection velocities.
In the text

	Fig. 3 Distribution of a, e, and i for lunar-ejected particles at the moment they reach the Earth. The symbols of a, e, and i denote the semi-major axis, the eccentricity, and the inclination of particles, which are measured in the ECI frame. Panel a: distribution of a. Panel b: distribution of e. Panel c: distribution of i.
In the text

	Fig. 4 Distribution of travel time of Earth impactors.
In the text

Fig. 5 Evolution of e, i, q, and Q for a 0.2 μm impactor from initial retrograde orbit, where e, i, q, and Q denote the eccentricity, the inclination, the perigee, and the apogee of the particle, which are measured in the geocentric lunar orbital frame. The dashed lines denote the time at which the particle leaves the Moon’s Hill sphere. Panel a: evolution of e. Panel b: evolution of i. Panel c: evolution of q and Q.

In the text

	Fig. 6 Fraction of solid angle subtended by particles per angular bin from the perspective of the Purple Mountain Observatory, at local midnight. The intervals of azimuth and elevation are 2° and 4°, respectively. The outer circle denotes the local horizon.
In the text

	Fig. 7 Fraction of solid angle subtended by particles per angular bin from the perspective of the Paranal Observatory of the European Southern Observatory, at local midnight.
In the text

	Fig. 8 Fraction of solid angle subtended by particles per angular bin from the perspective of the IceCube Neutrino Observatory, at local midnight.
In the text

	Fig. 9 Fraction of solid angle subtended by particles per angular bin from the perspective of the Helsinki Astronomical Observatory, at local midnight.
In the text

	Fig. 10 Fraction of solid angle subtended by particles per angular bin from the perspective of the Quito Astronomical Observatory, at local midnight.
In the text

Fig. 11 Geometric optical depth and number density of lunar-ejected particles relative to the Earth-Moon orbital plane, where the x-axis points to the ascending node of the Moon at the J2000 epoch, the z-axis is along the normal of the orbital plane (north), and the y-axis is determined by the right-handed rule. Panel a: normal geometric optical depth. Panel b: Azimuth-averaged number density in the ρ-z plane, where $\rho =\sqrt{x^2+y^2}$. The blue circle denotes Earth, and the red line denotes the orbit of Moon.

In the text

	Fig. 12 Fraction of solid angle subtended by particles in the ECI frame. The asterisks denote the bright stars, and the blue asterisks denote the stars that are more prone to being obscured by the lunar-ejected dust particles.
In the text