Abstract
NearEarth asteroid, Kamo’oalewa (469219), is one of a small number of known quasisatellites of Earth; it transitions between quasisatellite and horseshoe orbital states on centennial timescales, maintaining this dynamics over megayears. The similarity of its reflectance spectrum to lunar silicates and its Earthlike orbit both suggest that it originated from the lunar surface. Here we carry out numerical simulations of the dynamical evolution of particles launched from different locations on the lunar surface with a range of ejection velocities in order to assess the hypothesis that Kamo‘oalewa originated as a debrisfragment from a meteoroidal impact with the lunar surface. As these ejecta escape the EarthMoon environment, they face a dynamical barrier for entry into Earth’s coorbital space. However, a small fraction of launch conditions yields outcomes that are compatible with Kamo‘oalewa’s orbit. The most favored conditions are launch velocities slightly above the escape velocity from the trailing lunar hemisphere.
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Introduction
Small bodies in planetary systems can share the orbit of a massive planet in a longterm stable configuration by librating in the 1:1 meanmotion resonance^{1}; such configurations are referred to as coorbital motion. Examples of coorbital arrangements are known for many SolarSystem planets, the most ubiquitous being the large population of Trojan asteroids coorbiting with Jupiter. In the context of the idealized circular, restricted threebody problem (CR3BP), there are three main types of coorbital states: Trojan/tadpole (T), horseshoe (HS), and retrograde satellite/quasisatellite (QS)^{2}. The two cases of interest, horseshoe and quasisatellite, are shown in Fig. 1a, which are distinguished by the center of oscillation of their longitudes relative to Earth, of 180^{∘} and 0^{∘}, respectively.
Unlike the longterm stable population of Trojan asteroids coorbiting with Jupiter, most nearEarth asteroids (NEAs) have chaotic orbits with dynamical lifetimes much shorter than the age of the Solar System^{3}, and asteroids stably coorbiting with the Earth on such timescales are uncommon. An assessment of Earth’s coorbital companions shows a total population of at least twentyone objects, with two Trojantype, six in the QS state, and thirteen undergoing HS motion; all of these objects are in their coorbital states only temporarily, typically on less than decadal timescales^{4,5,6}. The recently discovered quasisatellite of the Earth, (469219) Kamo‘oalewa, is exceptional among the Earth’s coorbitals due to the longerterm persistence of its HS–QS transitions^{7,8,9,10}.
Kamo‘oalewa’s diameter is estimated to be 46–58 m^{11}, and its orbital elements are listed in Table 1, in which we observe that, although its semimajor axis is very close to Earth’s, its orbital eccentricity and inclination are not atypical of NEAs. Its ephemeris over a few centuries in the past and in the future, obtained from the Jet Propulsion Laboratory’s (JPL) Horizons web service, shows that the transition from HS to its current QS state occurred nearly a century ago; an event that will reverse in about 300 years when Kamo‘oalewa will again pass into a HS orbit (Fig. 1c).
Longterm numerical simulations indicate that these transitions will recur over hundreds of thousands or even millions of years^{9,10,12}. This can be contrasted with Earth’s firstknown recurrent quasisatellite, asteroid 2002 AA_{29}, whose future predicted QS state will last only for a few decades^{13,14}. Kamo‘oalewa’s close proximity to Earth and its unknown dynamical origin make it a scientifically compelling candidate for a future space mission^{15,16}.
Several hypotheses have been proposed for the origin of Kamo‘oalewa^{11,12}. Sharkey and colleagues measured its reflectance spectrum and found it to have an Ltype profile resembling lunar silicates^{11}, inconsistent with typical NEAs. These authors also concluded that Kamo‘oalewa is unlikely to be an artificial remnant from an earlier lunar mission. Its modest inclination could be indicative of a temporarily captured NEA, as is speculated for other planetary coorbitals^{17}. Its orbital eccentricity, however, is atypical of such captured coorbital states found in numerical simulations, which generally range between 0.3 (Venus crossing) and 0.6 (Mars crossing)^{18}. The other proposed scenarios are that Kamo‘oalewa might have originated in the EarthMoon system, either from a hitherto undiscovered quasistable population of Earth’s Trojans or as a lunar ejecta from a meteoroidal impact^{11}.
These latter scenarios for the provenance of Kamo’oalewa are at variance with prevailing theoretical models of nearEarth objects^{19,20} as these models assume only the main asteroid belt and comets as sources of NEAs. As a check, we employed the NEOMOD simulator^{20} and found that the latest model of NEAs does not account for Kamo’oalewalike orbits.
The focus of the present work is to examine the hypothesis that Kamo‘oalewa originated as lunar ejecta. We approach this by numerically simulating test particles (TPs) launched from the Moon’s surface and following their subsequent orbital evolution. We use a physically plausible range of launching speeds and directions and four representative launch locations (Fig. 2). The dynamical evolution of lunar ejecta has been previously investigated with numerical simulations^{21}. While those authors focused on determining whether lunar ejecta impact the Earth or Moon or escape into heliocentric orbits, our work focuses on determining whether such particles have dynamical pathways that lead to coorbital states. This is a more delicate question because, as we will see, such outcomes require statistically rare initial conditions (ICs); to our knowledge, this question has not been previously investigated.
Results
As in previous numerical investigations of lunar ejecta, a variety of dynamical behaviors were found as particles entered heliocentric orbits. In order to depict the global results graphically, we projected the orbital evolution of the particles onto the semimajor axis–eccentricity plane, as shown in Fig. 3.
An immediately observable feature in this diagram is that most of the launched particles evolve into orbital parameter regions traditionally demarcated as the Aten and Apollo regimes of the population of nearEarth objects. A similar result has been reported previously^{22}; it supports the suggestion that some of the members of the Aten and Apollo dynamical groups originate as lunar debris^{12}. The other noteworthy feature in Fig. 3 is the vertical structure of a low density of points around a = 1 AU. This is the coorbital region where we find the current orbit of Kamo‘oalewa and other HS–QS coorbital NEAs. The evident welldefined boundaries of this region show that there is a dynamical barrier between lunar ejecta and coorbital states but the finding of some outcomes in this region indicates that the barrier is somewhat porous, allowing a small fraction of lunar ejecta to evolve into and remain in coorbital states for varying periods of time.
We identified the coorbital outcomes by visual inspection of the time evolution of the particles that spend some time within the semimajor axis zone of 0.98–1.02 au. Overall, we found that 6.6% of all launched particles exhibited at least temporary coorbital motion, most as HS (5.8%) and some as HS–QS (0.8%). A particle had to perform at least one HS or QS oscillation to be considered temporarily in a coorbital state. A quantitative summary of the frequency for each dynamical outcome from each of the four launch sites is given in Table 2 along with the total collisions detected from each site. The trailing side produced the most coorbiters (both HS and QS), followed by leading side, and next by near side and far side which produced similar statistics.
Amongst all the initial conditions we simulated, some are more favorable for coorbital outcomes than others. This is illustrated in Fig. 4; the histogram in Fig. 4b shows the distribution of the initial launch speed of the cases that resulted in HS and HS–QS outcomes. Overall, the most favored launch velocity for HS outcomes is near the minimum of the sampled range (i.e., just above lunar escape velocity); for HS–QS outcomes (i.e., Kamo’oalewalike) the most favored initial launch speed is moderately higher, ~(4.0–4.4) km s^{−1}, in agreement with the estimates from the Section “Theoretical estimates”. In general, the total number of HS–QS outcomes decreases as the speed increases, discouraging exploration for larger values.
For additional detail, we examined the outcomes as a function of the radial and tangential components of the launch velocity (see Fig. 2 for an illustration of those directions), and made scatter plots of these velocity components for the four launch locations. In Fig. 4a, the initial launch velocity components from the lunar nearside are plotted as the yellow dots and those from the lunar farside are in gray. In Fig. 4b, the initial launch velocity components from the lunar trailingside are plotted as the yellow dots and those from the lunar leadingside are in gray. The HS and HS–QS outcomes are highlighted as the red and blue points. A clear asymmetry can be observed in these diagrams: most of the coorbital outcomes were launched with a negative tangential velocity (i.e., in the trailing direction of the Moon’s orbital motion). It is also evident that, out of the four representative launch sites considered, the trailing side is the most prolific in producing coorbiters. Additionally it can be noticed that most of the coorbitals produced from the leading side arise from the lower launch speeds, while for the other sites most of them arise from moderately larger launch speeds (>~3 km s^{−1}) (this will be shown to be due to the higher frequency of collisions for these conditions, as exposed in Fig. 5). We can also observe that for the larger launch speeds, in the range 4–6 km s^{−1}, coorbital outcomes are favored for launch directions in the radial or antiradial direction. These patterns in the outcomes are consistent with the theoretical expectations outlined in the Section “Theoretical estimates”.
It is perhaps noteworthy that we did not find any tadpoletype outcomes, that is, particles librating around just the L4 or the L5 Lagrange points. Other possible fates that were examined were collisions. Collisions with the Moon and the Earth were registered, most of them occurring at the lower launch speeds and within the first 100 years of their evolution. The statistics of the collision outcomes is shown in Fig. 5, in a scheme analogous to Fig. 4. That is, panels (a) and (b) show the scatter plots of the initial conditions that end in collisions and panel (c) plots a histogram of the frequency of collisions at different launch speeds. We observe a clear, rapid decay of collision outcomes for larger launch speeds. The distribution appearing in Fig. 5a, b, being concentrated at the lower speeds from near, far, and trailing side, accounts for the low frequency of coorbital outcomes under these conditions, as a particle that may have reached such a state would have collided before it could enter into a coorbital state.
Among the cases of HSQS coorbital outcomes observed in the simulations, most of them (around 66%) displayed only one transition or departed the QS state rapidly (before 1000 years), performing only one or two transitions. The orbits of interest are those whose HS–QS transitions recur persistently in a stable fashion for thousands of years, like Kamo‘oalewa. For the nine ICs that showcased such Kamo‘oalewalike dynamics (henceforth referred to as KL’s; see Fig. 3), the evolution was tracked further, for up to 100,000 years, or until they departed their coorbital states. Figure 6 shows an example KL outcome with persistent HS–QS transitions; this particle has recurrent residence times of 400–600 years in the QS state, inbetween shorter residence times in the HS state. For comparison, Kamo‘oalewa’s current time of residence in the QS state is ~400 years.
As previously noted, Kamo‘oalewa possesses a modest ecliptic inclination of about 8^{∘}. The orbital planes of most simulated ejecta particles were found to remain close to the ecliptic plane (typical inclinations ~ 1^{∘}–3^{∘}). This is because of our adopted simplification of considering initial launch conditions in a projected plane close to the ecliptic (see Fig. 2 and the associated description in the Section “Numerical model”). However, we did find some cases of KL outcomes in which the inclinations were driven up to higher values, reaching inclinations similar to Kamo‘oalewa’s. This can be seen in Fig. 7, where we plot the time evolution of the inclination of a simulated particle (KL2), as well as that of Kamo‘oalewa’s orbit. We observe episodic higher inclination states that persist for a few thousand years inbetween shortlived low inclination states. The higher inclination states are reached by a sequence of inclination jumps that occur at close approaches between the particle and Earth, and these jumps build up coherently over time spans of a few hundred years. In this figure, we also plot Kamo’oalewa’s inclination evolution over a 10,000 year time span, revealing similar inclination jumps at close approaches with Earth during its HS state as well as similar features when transitioning from HS to QS states, but with a boost in inclination rather than a decrease. These results demonstrate that Kamo’oalewa’s inclination could have arisen from a smaller initial inclination by means of kicks at close approaches during its HS state.
Discussion
It has been suggested that three small NEAs—2020 PN1, 2020 PP1, and 2020 KZ2, of estimated sizes in the range 10–50 m^{23,24,25}—may have the same provenance as Kamo‘oalewa due to their close orbital clustering and the similarities they exhibit in their orbital evolutions on timescales of a few thousand years^{5}. We have not investigated the orbital dynamics of these individual objects, but their resemblance to Kamo‘oalewa’s orbital elements implies that our results for Kamo‘oalewa could also be applicable to these objects’ origin.
The lunar ejecta hypothesis for the provenance of Kamo‘oalewa and other small Earth coorbitals can be tested for consistency with the lunar impact crater record and cratering mechanics. The lunar ejecta velocities (in excess of lunar escape speed, 2.4 km s^{−1}) needed to obtain the coorbital outcomes appear to be achievable in meteoroidal impacts on the Moon. Impacts on the Moon have typical impact speed of 22 km s^{−1} and as high as 55 km s^{−1}^{26,27}. Very small ejected debris particles may achieve comparable speeds, although the total fraction of such very high–velocity ejecta (solid or molten) is exceedingly small^{28,29}. Based on studies of lunar secondary craters, it is estimated that an escaping lunar ejecta fragment of size in the tens of meters would be expected only from relatively large impact craters, of diameter exceeding ~ 30 km^{30,31}. During the past ~1100 Ma of lunar history (the Copernican period in the lunar geological timescale), there were 44 impact craters of diameter exceeding 30 km^{32}, indicating that such large impacts occur at average intervals of about 25 Myr. The implication is that if Kamo‘oalewa is a lunar impact ejecta fragment, then it was launched from the lunar surface \({{{{{{{\mathcal{O}}}}}}}}(1{0}^{7})\) years ago. We leave to a separate study to investigate whether a lunar crater of appropriate size and age and geographic location can be consistent with the lunar ejecta hypothesis for the provenance of Kamo‘oalewa. If supported by such studies, Kamo‘oalewa would, to the best of our knowledge, be the first nearEarth asteroid to be recognized as a fragment of the Moon. It would be of great interest for cosmochemical study as a sample of ancient lunar material. The rarity of Kamo’oalewalike orbital outcomes (compared to Aten and Apollolike outcomes) in our simulations of escaping lunar ejecta suggests that many other lunar ejecta remain to be identified amongst the background population of nearEarth asteroids. This prediction is testable with nearinfrared reflectance spectra of very large numbers of NEAs that will be obtained by the forthcoming NearEarth Object Surveyor project^{33}.
Additional exploration of the orbital evolution of lunar ejecta is also warranted. Our numerical investigations reported here were limited in a number of ways, so it is useful to list some future directions of investigation. In the present study, we identified the mostfavorable launch velocities of lunar ejecta for Kamo‘oalewalike outcomes for initial conditions of the Solar System taken near the present epoch. Although in the Section “Numerical model” we invoked the Copernican principle that the present epoch is not “special," we do recognize that our results may have some sensitivity to the initial epoch. The most important limitation is due to Earth’s orbital eccentricity, which is time variable and undergoes excursions up to five times its current value on timescales of \({{{{{{{\mathcal{O}}}}}}}}(1{0}^{6})\) years. Consequently, Earth’s orbital velocity varies by ~2 km s^{−1}, an amount that is comparable to (or a significant fraction of) the launch velocities of escaping lunar ejecta. Therefore, this could influence the frequency of the coorbital outcomes of escaping lunar ejecta. Thus, sampling initial epochs when Earth’s eccentricity is different is needed to understand more comprehensively the statistics of coorbital outcomes of lunar ejecta. Sensitivity to epoch could also arise from lunar phase at launch (because the relative magnitude of solar perturbations on escaping lunar ejecta particles at full moon versus new moon phase is also a significant fraction of the lunar orbital velocity) and from the perturbations of Jupiter and other planets that would be slightly different at different epochs.
We would also like to understand the dynamical mechanism by which Kamo‘oalewa’s persistent HS–QS transitions occur. Of the three possible coorbital states, the QS state is the rarest found among small bodies in the Solar System. In the simple model of the planar, circular, restricted three body problem (PCR3BP), the intrinsic stability of nearly coplanar QS orbits has long been established^{34,35,36}. It has been linked to the existence of the family f of periodic orbits in the PCR3BP^{37}, and referred to as distant retrograde orbits (DROs) in applications to spacecraft navigation and mission design^{38,39}. In the spatial problem, vertical instabilities can arise and transitions between coorbital states are possible^{40,41,42,43,44}. In the regime of large eccentricity and inclination^{2}, has attributed such transitions to a secular drift of the asteroid’s perihelion and^{12} has suggested this as a mechanism that applies to Kamo‘oalewa. While the eventual escape from coorbital states may be linked to planetary secular perturbations^{45,46,47}, or to planetary close encounters^{48}, or to Yarkovskydriven migration^{10}, it is likely that the shorttime transport dynamics of Kamo’oalewa are governed by the invariantmanifold structure of the Lagrange points^{43,49,50}. For example, some authors attribute the entry and escape mechanisms of Kamo‘oalewa’s HS–QS transitions to such phasespace structures^{50}, but others invoke chaotic tangles of the Lagrange points to explain the dynamical mechanisms of capture into sticky QS orbits^{38}. Nevertheless, it is challenging to identify the specific phasespace structures responsible for the dynamical transport phenomena exhibited by Kamo‘oalewalike objects. More research is needed to understand the precise role of these manifolds on the dynamics of coorbital objects like Kamo‘oalewa, as well as on the wider NEA populations, and their implications for the asteroid impact hazard on our home planet^{44,51}.
The complex and nonlinear nature of the calculations performed leads to a large sensitivity to several conditions. For instance, initial conditions for objects in the Solar System were gathered from the JPL Horizons service, where masses and orbital elements are subject to updates and refinements. Further inconveniences arise from the fact that the orbital fates are classified based on visual inspection of 5000 year of orbital evolution of more than 10,000 simulated particles, so results are vulnerable to human error. Different results may also arise if the initial integration time of 5000 years is modified.
Conclusions
In our numerical simulations of the dynamical fates of lunar ejecta, we explored a representative range of ejecta launch conditions expected from large meteoroid impact events. The vast majority (more than 93%) of the launch conditions we considered resulted in ejecta reaching heliocentric orbits similar to the Aten and Apollo groups of NEAs, with no coorbital behavior detected; this is consistent with previous results^{21,22}. However, in a small minority (6.6%) of cases we detected the existence of pathways leading to coorbital states, most commonly horseshoe orbits, but also those resembling Kamo’oalewa’s; the latter exhibit persistent transitions between quasisatellite and horseshoe orbits. These minority outcome events have not been previously reported. The existence of these outcomes lends credence to the hypothesis that Kamo‘oalewa could indeed be lunar ejecta. The launch conditions most favored for such an outcome are those with launch velocities slightly above the lunar escape velocity and launch locations from the Moon’s trailing side. We also find that Kamo’oalewa’s inclination may have been boosted by close approaches with the Earth during its horseshoe state.
Methodology
Theoretical estimates
We begin with the observation that particles originating in the EarthMoon (EM) system that escape and evolve into Earthlike heliocentric orbits, including coorbital states such as horseshoe and quasisatellite types, would be those that escape with low relative velocity with respect to the EM barycenter. Here we make some estimates of the dynamical conditions of launch from the lunar surface that would favor outcomes with Earthlike heliocentric orbits.
For these estimates, we will take the Earth’s Hill sphere as the approximate boundary between geocentric and heliocentric space, and the lunar Hill sphere as the approximate boundary between selenocentric and geocentric space. The radius of the lunar Hill sphere is ~35 lunar radii:
and Earth’s Hill sphere radius is ~1% of Earth’s heliocentric orbit radius:
Here are the lunar mass, the mass of Earth + Moon, and the solar mass, respectively, are the lunar orbit radius and Earth’s heliocentric orbit radius, respectively (both approximated as circular orbits), and is the lunar radius.
We denote with v_{L} the launch velocity of a particle launched from the Moon’s surface; this is relative to the lunar barycenter. Particles launched with v_{L} exceeding the Moon’s escape speed, km s^{−1}, will reach the lunar Hill sphere boundary with a residual speed δv_{L} relative to the lunar barycenter. The magnitude of this residual velocity is estimated from the equation for conservation of energy in the lunar gravitational field, and is given by
For later reference, we observe that for a particle launched with , the residual velocity at the lunar Hill sphere radius is δv_{L} ≈ 0.4 km s^{−1}.
The velocity of an escaping lunar ejecta particle relative to the EarthMoon barycenter, v_{EM}, is found by adding (vectorially) the residual velocity δv_{L} to the lunar orbital velocity, ,
The Moon’s orbital velocity about the EarthMoon barycenter is km s^{−1}. To escape from the EarthMoon system, the magnitude of v_{EM} must exceed km s^{−1}. From Eq. (4), we see that this requires that δv_{L} must exceed 0.4 km s^{−1}. This minimum value is coincidentally the same as the residual velocity at the lunar Hill sphere of lunar ejecta launched with justthe lunar escape speed, that is, .
The magnitude of v_{EM} depends upon the location and speed of launch. We illustrate with two limiting cases. First consider lunar ejecta launched in the vertical direction from the apex of the lunar leading hemisphere. Such ejecta will reach the lunar Hill sphere boundary with residual velocity in a direction nearly parallel to the Moon’s orbital velocity. Consequently, they will get boosted by ~1 km s^{−1} (the lunar orbital velocity) to v_{EM} > 1.4 km s^{−1}, assuring escape from Earth’s Hill sphere. The second limiting case is that of lunar ejecta launched vertically from the apex of the trailing lunar hemisphere. Such ejecta will reach the lunar Hill sphere boundary with residual velocity approximately antiparallel to the Moon’s orbital velocity, so their v_{EM} will be lower than δv_{L} by ~ 1 km s^{−1}. In this case a residual velocity of magnitude δv_{L} > 2.4 km s^{−1} is needed in order to achieve v_{EM} > 1.4 km s^{−1}. From Eq. (3), this requirement implies a launch velocity v_{L} > 3.4 km s^{−1}. Ejecta from other locations and different launch directions will require a minimum launch speed in the range 2.4–3.4 km s^{−1} in order to leave the EarthMoon Hill sphere.
In the geocentric phase, the initial location of an escaping lunar ejecta particle is approximately at one lunar orbit distance, , and its velocity is v_{EM} relative to the EarthMoon barycenter. The ejecta will reach the EarthMoon Hill sphere boundary with a residual velocity, δv_{EM}, relative to the EarthMoon barycenter given by the energy conservation equation in geocentric space,
Taking v_{EM} = 1.4 km s^{−1} (the minimum required to achieve escape from geocentric space), escaping lunar ejecta will enter heliocentric space with a residual velocity (relative to the EarthMoon barycenter) of δv_{EM} = 0.7 km s^{−1}. This is a small fraction, ~0.023, of Earth’s heliocentric orbital velocity. Particles having δv_{EM} close to this minimum value will enter heliocentric space with the most Earthlike orbits, and would be good candidates for entering coorbital states such as the horseshoe or quasisatellite orbits.
For high speed lunar ejecta, those launched in a direction opposite to the lunar orbital velocity achieve lower residual velocities relative to the EarthMoon barycenter. This means that launch locations from the lunar trailing hemisphere (that is, the hemisphere opposite to the Moon’s orbital motion around Earth) would be more favorable for Earthcoorbital outcomes of escaping lunar ejecta. The circumstance that the Moon is in synchronous rotation with its orbital motion (and likely has been so for most of its history^{52,53}) means that the favorable launch location can be geographically constrained in this way for even ancient epochs of launch times.
The above estimates are based on patching together three different pointmass, twobody models (Moon + TP, Earth + TP, and finally Sun + TP). For the purposes of these simple estimates, we have also ignored the effect of the Moon’s rotation on the launch velocities of particles as well as the eccentricity of the lunar orbit and of Earth’s heliocentric orbit. In detail, the orbital evolution of escaping lunar ejecta particles that enter Earthlike heliocentric orbits is subject to strongly chaotic dynamics and is exceedingly sensitive to initial conditions, as is well known in the threebody problem, hence the need for the numerical approach that follows below. These theoretical estimates provide a guide for the initial conditions of the lunar ejecta that are to be explored with numerical simulations and a guide for the analysis of the results.
Numerical model
We explore the dynamical fates of lunar ejecta in a similar vein as has been done for satellite ejecta in the Saturnian system^{54}. Our dynamical model includes the eight major planets from Mercury to Neptune and the Moon, and we use the IAS15 integrator within REBOUND^{55}. The Direct predefined module in REBOUND was used to detect collisions with the massive bodies. An initial step size of 1.2 days was used and the step–size control parameter was set to its default value (ϵ = 10^{−9}; this assures machine precision for long time orbit propagations of 10^{10} orbital periods^{55}). The length of the main set of simulations was 5000 years; this is sufficiently long to explore the details of possible coorbital outcomes as a first study of the proofofconcept for the lunarejecta hypothesis for the origin of Kamo‘oalewa. In a second set of simulations, we extended the simulation time up to 100,000 years for those particles exhibiting Kamo‘oalewalike dynamical behavior. Running these simulations for much larger time spans is computationally prohibitive because, in order to detect the QS and HS dynamical states and transitions between such coorbital configurations, it is necessary to have high cadence outputs (~1 output per year); this places high demands on data storage.
The initial conditions for the planets are obtained from the JPL Horizons system at epoch J2452996, i.e., 22 December 2003. In this initial exploration, we adopt the Copernican principle^{56,57} that the current time is not special, and is not unrepresentative of lunar ejecta launch conditions at any random time in the geologically recent past. This assumption can be tested in the future by exploring different initial epochs that sample different initial conditions of the planets—especially of Earth’s eccentricity—on secular timescales.
The initial conditions for the test particles are generated through three parameters: the angle θ_{1} between the line joining the center of the Moon to the launch site on the lunar surface and the line joining the center of the Moon to center of Earth, the angle θ_{2} between the launch velocity vector and the local normal vector at the launch site on the lunar surface, and the speed of launch v_{L}. For simplicity, we consider a two dimensional projection of the Moon’s surface onto the ecliptic, illustrated in Fig. 2. Accordingly, the position of the TP is completely specified by θ_{1} (and the Moon’s radius), while its initial velocity (relative to the lunar surface) is determined by the magnitude and direction of the specified relative velocity (v_{L} and θ_{2}, respectively); according to the projection made, this relative velocity has no component perpendicular to the ecliptic plane. The angles θ_{1} and θ_{2} range from (0^{∘}, 360^{∘}) and (−90^{∘}, 90^{∘}), respectively. The values of v_{L} were chosen between 2.4 and 6.0 km s^{−1}, with the lower bound corresponding to the lunarescape speed and the upper bound being the limit of ejection velocities reported in numerical simulation studies of lunar cratering events^{58}. It should also be noted that the frequency of ejection velocities decreases rapidly as v_{L} increases^{58}, discouraging the exploration of larger values of v_{L}.
Following the guidance from the Section “Theoretical estimates”, four launch sites were sampled on the Moon’s surface. These four sites are representative of each of the four hemispheres of the Moon: the nearside, farside, leading side, and trailing side. The locations of these are shown in Fig. 2. At each location, the launch speed was varied systematically from 2.4 to 6.0 km s^{−1} (in increments of 0.1 km s^{−1}), and, for each speed, 100 particles were launched with different angles θ_{2} (uniformly chosen along −90^{∘} and 90^{∘}). In total, we launched 14,800 test particles.
In the simulations, we monitored for collisions with all the massive bodies. In order to identify coorbital outcomes, we visually examined the time series of a and Δλ. Rather than examining the time series of all launched particles, this task is made easier by first projecting the evolution in the (a, e) plane and identifying those particles that appear in a rather sparsely populated, narrow vertical zone in the semimajor axis range 0.98 − 1.02 au, as explained further in the “Results” section.
Data availability
Outcomes of the numerical simulations presented in this paper are available at this publicly accessible permanent repository: https://doi.org/10.5281/zenodo.8339513.
Code availability
REBOUND can be freely downloaded from the developers’ webpage https://rebound.readthedocs.io. Codes and scripts used for the numerical simulations may be requested to the corresponding author.
References
Murray, C. D. & Dermott, S. F. Solar System Dynamics (Cambridge University Press, Cambridge, 1999).
Namouni, F., Christou, A. A. & Murray, C. D. Coorbital dynamics at large eccentricity and inclination. Phys. Rev. Lett. 83, 2506–2509 (1999).
Gladman, P., Michel, P. & Froeschlé, C. The NearEarth object population. Icarus 146, 176–189 (2000).
Kaplan, M. & Cengiz, S. Horseshoe coorbitals of Earth: current population and new candidates. Mon. Not. R. Astron. Soc. 496, 4420–4432 (2020).
de la Fuente Marcos, C. & de la Fuente Marcos, R. Using Mars coorbitals to estimate the importance of rotationinduced YORP breakup events in Earth coorbital space. Mon. Not. R. Astron. Soc. 501, 6007–6025 (2021).
Di Ruzza, S., Pousse, A. & Alessi, E. M. On the coorbital asteroids in the solar system: mediumterm timescale analysis of the quasicoplanar objects. Icarus 390, 115330–20 (2023).
de la Fuente Marcos, C. & de la Fuente Marcos, R. Asteroid 2014 OL_{339}: yet another Earth quasisatellite. Mon. Not. R. Astron. Soc. 445, 2985–2994 (2014).
Dermawan, B. Temporal Earth coorbital types of asteroid 2016 HO_{3}. J. Phys. Conf. Ser. 1523, 012019–4 (2019).
Rezky, M. & Soegiartini, E. The orbital dynamics of asteroid 469219 Kamo‘oalewa. J. Phys. Conf. Ser. 1523, 012019–012016 (2020).
Fenucci, M. & Novaković, B. The role of the Yarkovsky effect in the longterm dynamics of asteroid (469219) Kamo‘oalewa. Astron. J. 162, 227–211 (2021).
Sharkey, B. N. L. et al. Characterizing Earth quasisatellite (469219) 2016 HO_{3} Kamo‘oalewa. Nature Commun. Earth Environ. 2, 231–237 (2021).
de la Fuente Marcos, C. & de la Fuente Marcos, R. Asteroid (469219) 2016 HO_{3}, the smallest and closest Earth quasisatellite. Mon. Not. R. Astron. Soc. 462, 3441–3456 (2016).
Brasser, R. et al. Transient coorbital asteroids. Icarus 171, 102–109 (2004).
Wajer, P. 2002 AA_{29}: Earth’s recurrent quasisatellite. Icarus 200, 147–153 (2009).
Venigalla, C. et al. NearEarth Asteroid Characterization and Observation (NEACO) mission to asteroid (469219) 2016 HO_{3}. J. Spacecr. Rockets 56, 1121–1136 (2019).
Jin, W. T. et al. A simulated global GM estimate of the asteroid 469219 Kamo‘oalewa for China’s future asteroid mission. Mon. Not. R. Astron. Soc. 493, 4012–4021 (2020).
de la Fuente Marcos, C. & de la Fuente Marcos, R. (309239) 2007 RW_{10}: a large temporary quasisatellite of Neptune. Astron. Astrophys. 545, 9–4 (2012).
Morais, M. H. M. & Morbidelli, A. The population of nearEarth asteroids in coorbital motion with the Earth. Icarus 160, 1–9 (2002).
Granvik, M. et al. Debiased orbit and absolutemagnitude distributions for nearEarth objects. Icarus 312, 181–207 (2018).
Nesvorný, D. et al. NEOMOD: a new orbital distribution model for nearearth objects. Astron. J. 166, 55 (2023).
Gladman, B. J., Burns, J. A., Duncan, M. J. & Levison, H. F. The dynamical evolution of lunar impact ejecta. Icarus 118, 302–321 (1995).
Bottke Jr, W. F. et al. Origin of the Spacewatch small Earthapproaching asteroids. Icarus 122, 406–427 (1996).
Small Body Database Lookup, 2020 PN1, accessed 30 June 2023, https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=54050997.
Small Body Database Lookup, 2020 PP1, accessed 30 June 2023, https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2020.
Small Body Database Lookup, 2020 KZ2, accessed 30 June 2023, https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html#/?sstr=2020.
Ito, T. & Malhotra, R. Asymmetric impacts of nearEarth asteroids on the Moon. Astron. Astrophys. 519, 63–9 (2010).
Gallant, J., Gladman, B. & Ćuk, M. Current bombardment of the EarthMoon system: emphasis on cratering asymmetries. Icarus 202, 371–382 (2009).
Liu, T., Luther, R., Manske, L. & Wünnemann, K. Melt production and ejection at lunar intermediatesized impact craters: where is the molten material deposited? In European Planetary Science Congress, pp. 2022–402 (2022).
Melosh, H. J. Highvelocity solid ejecta fragments from hypervelocity impacts. Int. J. Impact Eng. 5, 483–492 (1987). Hypervelocity Impact Proceedings of the 1986 Symposium.
Singer, K. N., Jolliff, B. L. & McKinnon, W. B. Lunar secondary craters and estimated ejecta block sizes reveal a scaledependent fragmentation trend. J. Geophys. Res. Planets 125, 2019–00631327 (2020).
Melosh, H. J. An Empirical Function Linking Impact Ejecta Fragment Size and Velocity. In 51st Annual Lunar and Planetary Science Conference. Lunar and Planetary Science Conference, p. 2587 (2020).
Stöffler, D. et al. Cratering history and lunar chronology. Rev. Mineral. Geochem. 60, 519–596 (2006).
NearEarth Object Surveyor Mission (2022), accessed 30 June 2023, https://neos.arizona.edu/.
Shen, Y. & Tremaine, S. Stability analysis of Earth coorbital objects. Astron. J. 136, 2453–2467 (2008).
Pousse, A., Robutel, P. & Vienne, A. On the coorbital motion in the planar restricted threebody problem: the quasisatellite motion revisited. Celest. Mech. Dyn. Astr. 128, 383–407 (2017).
Voyatzis, G. & Antoniadou, K. I. On quasisatellite periodic motion in asteroid and planetary dynamics. Celest. Mech. Dyn. Astr. 130, 59–18 (2018).
Hénon, M. Numerical exploration of the restricted problem. V. Hill’s case: periodic orbits and their stability. Astron. Astrophys. 1, 223–238 (1969).
Scott, C. J. & Spencer, D. B. Calculating transfer families to periodic distant retrograde orbits using differential correction. J. Guid. Control. Dyn. 33, 1592–1605 (2010).
Lara, M. Nonlinear librations of distant retrograde orbits: a perturbative approach—the Hill problem case. Nonlinear Dyn. 93, 2019–2038 (2018).
Mikkola, S. et al. Stability limits for the quasisatellite orbit. Mon. Not. R. Astron. Soc. 369, 15–24 (2006).
Sidorenko, V. V., Neishtadt, A. I., Artemyev, A. V. & Zelenyi, L. M. Quasisatellite orbits in the general context of dynamics in the 1:1 mean motion resonance: perturbative treatment. Celest. Mech. Dyn. Astr. 120, 131–162 (2014).
Morais, M. H. M. & Namouni, F. Periodic orbits of the retrograde coorbital problem. Mon. Not. R. Astron. Soc. 490, 3799–3805 (2018).
Oshima, K. The roles of L_{4} and L_{5} axial orbits in transport among coorbital orbits. Mon. Not. R. Astron. Soc. 480, 2945–2952 (2018).
Qi, Y. & Qiao, D. Stability analysis of Earth coorbital objects. Astron. J. 163, 211–14 (2022).
Wajer, P. Dynamical evolution of Earth’s quasi satellites: 2004 GU9 and 2006 FV35. Icarus 209, 488–493 (2010).
Christou, A. A. & Asher, D. J. A longlived horseshoe companion to the Earth. Mon. Not. R. Astron. Soc. 414, 2965–2969 (2011).
Connors, M. A Kozairesonating Earth quasisatellite. Mon. Not. R. Astron. Soc. 437, 85–89 (2014).
Christou, A. A. & Georgakarakos, N. Longterm dynamical survival of deep Earth coorbitals. Mon. Not. R. Astron. Soc. 507, 1640–1650 (2021).
Barrabé, E. & Ollè, M. Invariant manifolds of L_{3} and horseshoe motion in the restricted threebody problem. Nonlinearity 19, 2065–2089 (2008).
Qi, Y. & Qiao, D. Coorbital transition of 2016 HO_{3}. Astrodyn. (in press, 2022).
Todorović, N., Wu, D. & Rosengren, A. J. The arches of chaos in the Solar System. Sci. Adv. 6, 1313–6 (2020).
Wieczorek, M. A. & Le Feuvre, M. Did a large impact reorient the moon? Icarus 200, 358–366 (2009).
Aharonson, O., Goldreich, P. & Sari, R. Why do we see the man in the Moon? Icarus 219, 241–243 (2012).
Dobrovolskis, A. R., Alvarellos, J. L., Zahnle, K. J. & Lissauer, J. J. Exchange of ejecta between Telesto and Calypso: tadpoles, horseshoes, and passing orbits. Icarus 210, 436–445 (2010).
Rein, H. & Spiegel, D. S. IAS15: a fast, adaptive, highorder integrator for gravitational dynamics, accurate to machine precision over a billion orbits. Mon. Not. R. Astron. Soc. 446, 1424–1437 (2014).
Peacock, J. A. Cosmological Physics (1999).
Gott Jr, I. Implications of the Copernican principle for our future prospects. Nature 363, 315–319 (1993).
Artemieva, N. A. & Shuvalov, V. V. Numerical simulation of highvelocity impact ejecta following falls of comets and asteroids onto the Moon. Sol. Syst. Res. 42, 329–334 (2008).
Acknowledgements
We acknowledge an allocation of computer time from the UA Research Computing High Performance Computing (HPC) at the University of Arizona. The citations in this paper have made use of NASA’s Astrophysics Data System Bibliographic Services. R.M. additionally acknowledges research support from the program “Alien Earths” (supported by the National Aeronautics and Space Administration under agreement No. 80NSSC21K0593) for NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate, and from the Marshall Foundation of Tucson, AZ. A.R. acknowledges support by the Air Force Office of Scientific Research (AFOSR) under Grant No. FA95502110191.
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J.D.C.C. wrote the codes and scripts used for the numerical simulations, conducted the simulations, data processing, and data analysis. R.M. composed the theoretical estimates and carried out related dynamical calculations. R.M. and A.J.R. designed the research project. All authors discussed and critiqued interim and final results and contributed to writing the manuscript.
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CastroCisneros, J.D., Malhotra, R. & Rosengren, A.J. Lunar ejecta origin of nearEarth asteroid Kamo’oalewa is compatible with rare orbital pathways. Commun Earth Environ 4, 372 (2023). https://doi.org/10.1038/s4324702301031w
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DOI: https://doi.org/10.1038/s4324702301031w
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