1 Introduction

As outlined in previous work (Armstrong et al. 2002; Gutiérrez 2001, 2002), the surface of the Moon, lacking a substantial atmosphere, hydrology, or geological activity over long periods of time, may represent an ideal location to search for surviving fragments of material blasted off of Earth, Venus, and Mars in the past 4 billion years during giant impact events. Up until this point, we have only managed to calculate bulk transfer rates to the moon, and these results have lacked any ability to resolve likely impact locations, angles, or velocities to aide in the search and recovery of terrestrial meteorites on the Moon. It is the goal of this paper, as an aide to future lunar exploration, to provide maps of likely impact locations as well as determine the best places to find the least modified of the terrestrial meteorites.

Section 2 describes the numerical models used to perform the simulations and produce the maps, distribution of angles, and distribution of velocities. In Sect. 3, we present the results and compare these to previous work. Finally, in Sect. 4, we discuss the results and present the best locations for locating terrestrial material on the Moon.

2 Numerical Model of Direct Transfer

For this work, we used the hybrid symplectic Bulirsch-Stoer algorithm in the planetary dynamics simulation software Mercury (Chambers and Murison 2000; Chambers 1999). Since the purpose of these simulations is to resolve collisions of ejecta on the surface of the Moon, we integrated Earth, the Moon, and the Sun as massive bodies in the simulations, with the remaining particles treated as massless for the purposes of self-interaction.

To account for the fact that the Moon has receded from the Earth over time, we ran simulations over a range of Earth-Moon distances. As outlined in Zharkov (2000) and discussed in Armstrong et al. (2002), we use an earth-moon distance of \(21 \,R_{\oplus}\) as the starting point for the simulations, and ran nine cases up to the current value of approximately \(61 \,R_{\oplus}\). The initial conditions for the simulations and the total number of particles for each are listed in Tables 1 and 2.

Table 1 The simulation time, initial timestep, and initial conditions of Earth and the Moon for the simulations
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Table 2 Details of the simulations
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Each simulation consists of 1,000 particles at a given velocity between Earth’s escape velocity and the minimum velocity needed to reach the lunar orbit at that distance, given by

$$ v_{\rm min} = v_{\rm esc} \left(1 - \frac{1}{\eta}\right)^{\frac{1}{2}}, $$
(1)

where η is the earth-moon distance in Earth radii. The velocity bins are broken up in 1 m s−1 intervals such that the total number of simulations are determined by the width of the velocity range. The simulations were run on Weber State University’s 128 node distributed computing cluster. The results for each lunar distance were concatenated after all simulations were completed.

The Mercury software records close encounters between particles when they are within a specific number of Hill radii, while the collision algorithm detects when actual impacts occur. For the sake of the simulation, the position and velocity of the colliding particles are recorded and the particles are combined for the remainder of the simulation. In our case, collisions between test particles were ignored, and lunar impacts recorded. After the simulations were completed, we used the center of mass position of both the particle and the moon, along with the lunar radius, to compute the latitude and longitude of the impact on the moon. For these simulations, we assumed that the lunar shape was frozen in during the late heavy bombardment (Zharkov 2000), and that tidal locking occurred prior to 3.9 billion years ago. In addition to recording the latitude and longitude of impact, relative to the sub-earth point, we computed the impact angle and velocity vector for each collision.

The previous numerical work for orbital transfer assumed a spherically symmetric distribution of initial particle positions with the velocity vectors aligned with the center of Earth. This simulates an impact event, and, in the case of direct transfer, allows the moon to interact with as many particles as possible. However, since the goal is to record as many lunar collisions as possible, we restricted the initial conditions for the particles to an annulus evenly distributed about Earth under the sub-lunar point just wide enough to ensure interaction with the moon. As the total transfer efficiencies will depend on all the material leaving the surface, the transfer fractions must be corrected for the bias introduced in the initial conditions.

To calculate the transfer efficiencies, we need to account for the ratio in area of the annulus used in the simulation compared to the spherical approximation used in previous work. The initial area covered by the simulation particles is

$$ A_i = \int\limits^{2\pi}_{0} \int\limits^{\phi_2}_{\phi_1} R_{\oplus}^2 sin\phi d\theta d\phi = 2 \pi R_{\oplus}^2 \int\limits^{\phi_2}_{\phi_1} sin\phi d\phi $$
(2)

where \(R_{\oplus}\) is the radius of Earth,

$$ \phi_1 = \frac{\pi}{2} - \frac{\pi}{128}, $$
(3)

and

$$ \phi_2 = \frac{\pi}{2} + \frac{\pi}{128}. $$
(4)

Dividing this area by the area of Earth’s surface, A e , we have the area correction parameter

$$ \alpha = \frac{A_i}{A_e} = 2.45\times10^{-2}. $$
(5)

This parameter is used to correct the ratio of impacts to total number of particles in the simulations to get an area adjusted impact fraction that can be compared to previous studies and represents an estimate of the actual transfer efficiency of Earth ejecta to the lunar surface.

Table 3 shows the concatenated results from all nine simulations, including the total number of impacts during the simulations for all initial velocities as well as the area adjusted impact fraction for the simulation. To make a direct comparison to mass fraction transferred to the moon, F direct from Armstrong et al. (2002), we follow the same procedure outlined in that paper to compute the mass fraction that escapes Earth for a given range of velocities, and multiply that by the transfer efficiencies calculated from the N-body simulations. Table 4 compares the results from this simulation and the previous analytical method.

Table 3 Impact transfer fractions as a function of Earth-Moon distance
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Table 4 Comparison of the new estimates for the fraction of the impactor mass transferred to the Moon (F direct) and those from the previous analytical calculation
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3 Results

Tables 3 and 4 show that, out of nearly 1.4 million simulated test particles, 4678 impacted the moon, giving an average area-adjusted impact fraction of 8.2 × 10−5, with significantly higher transfer efficiencies when the moon is closer to the Earth. Accounting for the fraction of mass that leaves Earth from the original impact shows that, on average, F direct = 6.4 × 10−7, but was as high as 2.0 × 10−6 during the late heavy bombardment when Earth was subject to the highest incident of impactors. According to these results, however, the analytical method of Armstrong et al. (2002) overestimated the transfer rates by a factor of 3.3–5.5. Since the direct transfer in Armstrong et al. (2002) was cited as the primary transfer method of material between Earth and the Moon, this reduces the previously estimated globally-averaged concentrations of lunar material on the moon from 7 ppm to between 1–2 ppm.

However, these new simulations show that the distribution of impacts on the moon are far from uniform. Figure 1 shows the statistics of the recorded impacts on the moon. The top plot shows the fraction of impacts as a function of time on a log-log plot, showing the majority of the impacts occur early on in the simulation (the out-bound and return trip for most particles with initial velocities less than the escape velocity). The middle panel shows the fraction of impactors as a function of latitude, adjusted for planetary surface area, showing that an equivalent number of particles hits each latitude region. The final panel shows the distribution of particles as a function of longitude, showing the largest concentration on the far side of the moon. This is likely due to the fact that the far side effectively gets two chances to collect terrestrial meteorites, once during the outbound trip, and once as the material falls back to Earth.

Fig. 1
figure 1

Histograms of the impact time (top panel), impact latitude (middle panel) and impact longitude (bottom panel). The impact fraction as a function of latitude has been corrected for the change in surface area at higher latitudes

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Figure 2 shows a sinusoidal projection of the lunar surface with the locations of the impacts indicated. The size of the dots are the vertical component of the total impact speed, which is the quadrature sum of the particle velocity and the lunar orbital velocity. Thus, direct impacts are indicated by the largest symbols, while oblique impacts are indicated by the smaller symbols. As expected, we see the increased number of impacts on the far side (the edges of the map) as well as the increase in direct hits on the leading edge of the moon. Figure 3, the near side of the moon and Fig. 4, the far side of the moon, show these effects in more detail. For reference, the locations of the Apollo landing sites are indicated on both Figs. 2 and 3.

Fig. 2
figure 2

The distribution of impact locations on the moon, sinusoidal projection. Each circle represents an impact event with the circle’s size proportional to the vertical impact speed, according to the legend. The largest impact velocities are concentrated on the leading edge of the moon

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Fig. 3
figure 3

The near side geological distribution of impact sites. The highest density of impacts is at both poles, which also correspond to the largest impact velocities

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Fig. 4
figure 4

The far side geographical distribution of impact sites. The far side contains the most number of impacts with vertical impact speeds below 1.0 km s−1

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With nearly 5000 recorded impacts during the course of the simulations, it is possible to develop some statistics quantifying the best locations for searching for terrestrial material. To this end, we broke the lunar surface into 18 longitude bins and 18 latitude bins. For each 20° by 10° bin, we tabulated the total number of impacts occurring during all simulations to compute the surface density of impacts,

$$ \sigma_{i,j} = \frac{N_{i,j}}{A_{i,j}}, $$
(6)

where A i,j is the surface area of the latitude/longitude bin. This was normalized by the average impact surface density, < σ > , to get the normalized impact surface density,

$$ \Upsigma_{i,j} = \frac{\sigma_{i,j}}{\left<\sigma\right>}. $$
(7)

This number gives how much a given region is above or below the average impact surface density. \(\Upsigma\) ranges from a factor 0.04–8.4 over the entire surface of the moon.

Perhaps more interesting are the areas that have both the highest impact surface density, but also the lowest impact velocities. For each bin, we normalized the impact surface density by the average vertical velocity of those impacts, < v > ,

$$ \Upsigma_{v} = \frac{\Upsigma}{\left<v\right>}. $$
(8)

Table 5 shows the results for the most promising regions with the highest impact surface density, \(\Upsigma\), and the highest velocity-normalized impact surface density, \(\Upsigma_{v}\). The region centered on −50 W, 5 S was the least promising location, receiving only one impact, which is 0.04 fewer impacts per square kilometer than average. The most promising location, centered on 50 W, 85 S, received 8.4 times more impacts per square kilometer than average.

Table 5 Most promising sites for recovery of terrestrial material
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Now that we have resolved the impact locations as well as the impact angles and velocities, we can address unanswered questions from previous work. As detailed in Armstrong et al. (2002) and, more recently in Crawford et al. (2008), the survivability of biomarkers in terrestrial material depends on the peak pressure experienced by the material during impact. More direct impacts will hit with the full force of the impact velocity, while more oblique impacts will experience lower peak pressures and temperatures. Figure 5 shows a histogram of the impact angles for the impacts in all the simulations. We see a bimodal distribution, indicating that nearly 20% of the impacts occur nearly head on (>70°) and almost 50% hit at oblique angles (<20°).

Fig. 5
figure 5

The distribution of impact angles. The distribution is bimodal, indicating direct hits (>80°) and glancing blows (<20°)

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Figure 6 shows the histogram for the impact velocities. The solid line is the magnitude of the impact vector, indicating a clear minimum near the lunar escape speed of 2.4  km  s−1. However, when the angles are taken into account by computing the vertical impact speed, a new distribution emerges (dashed line). This shows the vertical component, which governs the peak pressures and temperatures in the impactor can be quite low, with nearly 10% of all impacts having a vertical impact speed of 0.1 km s−1 or less. As the peak pressures occur on the leading edge, with pressures elsewhere in the impactor being less severe (Crawford et al. 2008), these oblique impacts can potentially produce lightly shocked material. Table 6 updates the analytical estimates in Table 3 of (Armstrong et al. 2002), outlining the peak pressures associated with each velocity and the probability of impacts occurring less than the stated velocity. According to these results, while there are similar numbers of impactors with the lowest vertical velocities, Armstrong et al. (2002) systematically underestimated the fraction of higher velocity impactors. This is likely due to the fact that the analytical method could not take into account the bimodal distribution of impact angles shown in Fig. 5.

Fig. 6
figure 6

The distribution of impact velocities. The solid line represents the magnitude of the impact velocities and the dotted line is the vertical components of the impact velocity

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Table 6 Vertical impact speeds, associated probabilities, and peak pressures
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It needs to be emphasized that the peak pressures estimated by the analytical method of Armstrong et al. (2002), listed in Table 6, while in the ballpark, are systematically overestimated for the direct impacts, and underestimated for the oblique impacts. Crawford et al. (2008) performed more robust hydrodynamical simulations for direct and oblique impacts. The median peak pressure for a direct impact in basalt, for a total impact velocity of 2.5 km  s−1, is 3.7 GPa, and as low as 2.5 GPa on the trailing edge of the impactor. The median peak pressure goes as low as 1.9 GPa for a 20° impact, with the trailing edge experiencing pressures as low as 0.8 GPa. Given that nearly 50% of all the impacts in these simulations have impact angles of 20°, the chances for weakly shocked impactors is high.

4 Discussion

The results of this work can be summarized as follows:

  1. 1.

    Significant amounts of terrestrial material can be transferred between Earth and the Moon. While previous work has overestimated the global average by a factor of 3.3–5.5, specific regions on the Moon can be enhanced by a factor of 8.4 over the new globally averaged estimate. With a previous estimate of 7 ppm and 200  kg km−2, this puts the globally averaged concentration of terrestrial material between 1–2 ppm, or 36–61 kg km−2. However, some regions on the moon may have as much as 11-18 ppm, or 300–510 kg km−2. Conversely, some regions may have significantly less material, as little as 0.05–0.08 ppm (1.4–2.4 kg km−2). According to these results, site selection is an essential factor to successfully locating terrestrial material on the moon. It should be noted that all of the Apollo landing sites are located on low surface density regions for terrestrial meteorite impacts.

  2. 2.

    There is a strong bimodal distribution in impact angles, suggesting that a significant fraction of the impacts can occur at low impact angles and low vertical impact speeds. Thus, it is likely that the surface of the Moon contains a relatively large fraction of weakly shocked terrestrial material.

  3. 3.

    The region of the moon at 50 W, 85 S represents the best location to search for terrestrial material on the Moon. This has the advantage of being on the near side, and would be an excellent target for a near-term lunar mission. However, other regions at 170 W, 85 S and 130 W, 85 S are coincident with the extension of the South Pole/Aiken basin on the far side of the moon, a likely target for sample return in the near future.

An unanswered question remains: what is the distribution of meteorite fragments we should expect on the Moon? Wells et al. (2003) outline work that indicates fragments leaving Earth can be up to 6 m in size. The effect of lunar impact, even at relative low velocities, on the terrestrial material is largely unknown, though biologically interesting samples such as seeds have been shown to survive hypervelocity impacts in the laboratory (Levoci et al. 2009; Jerling et al. 2008; Bowden et al. 2008).

Depending on the fragmentation of the impactors, these simulations show that regions on the Moon with the highest concentrations could have as many as 300–500 10-cm diameter basaltic meteorites per square km if large fragments are common, or the entire sample could be pulverized and mixed into the lunar regolith. Large samples may be relatively easy for humans or rovers pick out, like finding meteorites on the surface of Mars (Schröder et al. 2008; Johnson et al. 2010), while small fragments mixed into the regolith may lend themselves to detection by micro-imaging spectrometers (Nunez et al. 2009; Nuñez et al. 2010)

Another significant challenge is identifying terrestrial material on the Moon. Mineralogy and composition aside, Gutiérrez (2002) has suggested that a fusion crust created when the material left Earth’s atmosphere might remain intact. Earth’s hallmark–water and hydrated minerals–may become a difficult marker, since evidence for indigenous water and hydrated materials on the Moon is mounting (Teodoro et al. 2010; Sridharan et al. 2010; Spudis et al. 2010; Pieters et al. 2009; Sunshine et al. 2009; Clark 2009). However, any hydrated material on the Moon–indigenous or exogenous–is of scientific interest, and this work supports the notion that at least some of the exogenous material on the Moon is of terrestrial origin.