1 Introduction

Impact crater formation is one of the main processes that formed the appearance of Solar system bodies throughout the time of its existence. On the surface of the Moon there are traces of collisions with a variety of impactors, such as asteroids, meteorites and comets, from the heavy bombardment (3.9 billion years ago) till the present day. Asteroids (this rare event occurred with a greater frequency in the past), meteorites, and comets can collide with the lunar surface. The size and form of a crater depend on many factors: velocity and angle of impact, impactor characteristics (such as size and density), and target properties. The dependencies of crater characteristics on impactor and target properties were studied in many researches (Davison et al. 2011; Holsapple and Housen 2007; Melosh 1989; Melosh and Pierazzo 2000; O’Keefe and Ahrens 1982; Schmidt and Housen 1987; Yue et al. 2013). By studying crater characteristics, it is possible to determine the nature of the body that formed the crater. However, since impact craters are modified throughout the time of their existence (destroyed by impacts with other bodies, covered by ejecta from other craters, collapse), obtaining initial impact parameters is a complicated task. Therefore, any peculiarities of the crater that can provide additional information regarding certain characteristics of the impact that formed the crater, simplify the task, albeit without providing a single accurate solution. In addition, recent studies found that remote craters on the Moon and other bodies could contain not only the substance of the target, but also that of the impactor (Bland et al. 2008; Melosh and Pierazzo 2000; Yue et al. 2013). Finding such craters on the lunar surface can usher a new era in the use of our satellite’s resources.

2 Background

2.1 Types and Characteristics of Impactors that Collide with the Moon

At present, the flux of interplanetary substance to the Moon is estimated at (2 ± 1) × 106 kg/h (Love and Brownlee 1993). Meteorites are divided into three main types: stony (subdivided into two groups: chondrites and achondrites), stony–iron, and iron–nickel. Stony meteorites are the most common ones, amounting to 92 % of all meteorites and to 85 % of meteorites found on Earth. The composition of chondrites includes silicate chondrules, mostly of spherical or elliptic form. Achondrites lack such inclusions. Most (92.7 %) stony meteorites are chondrites. Several types of chondrites are distinguished based on the oxidation level of their iron: enstatite chondrites, ordinary chondrites, and carbonaceous chondrites. Ordinary chondrites are encountered most frequently among stony meteorites. Their composition includes magnesium and iron silicates (minerals that contain olivine, pyroxene, plagioclase) and inclusions of metallic iron and nickel. It is assumed that parent bodies of ordinary chondrites are asteroids of the main asteroid belt.

Carbonaceous chondrites contain carbon compounds, including carbonates. The composition of carbonaceous chondrites includes large amounts of iron and hydrosilicates. Graphite and carbon dust, as well as some organic compounds, give meteorites a dark shade. Carbonaceous chondrites contain the most volatile compounds, such as O, H, S, and H2O.

Enstatite chondrites are rare (2 % of the total number of chondrites). Unbound iron is the main element of enstatite chondrites. They also contain enstatite, which named the entire group. Enstatite meteorites contain the smallest amount of volatile compounds, compared with other chondrites.

Achondrites contain large amounts of olivine and pyroxene; their composition is similar to that of lunar and terrestrial basalts. Achondrites are poor in iron and siderophile elements. It is assumed that achondrites are debris, ejected from planets and satellites after collisions with large bodies, which had differentiated into a crust, mantle and core. In particular, the Vesta and other differentiated asteroids is considered the parent body of achondrites (Gupta and Sahijpal 2010).

According to different estimations, stony–iron meteorites account for 1.5–2 % of the total number and 3.5–5 % of the total mass of meteorites found on Earth. Stony–iron meteorites contain an approximately equal amount of iron and silicates (including olivine). For iron meteorites, these amounts are 6 and 10 %, respectively. Iron meteorites are mostly of iron–nickel composition.

The density of meteorites changes, depending on their type. The density of ordinary chondrites varies from 2.1 to 3.7 g/cm3 (Britt and Consolmagno 2004); the density of carbonaceous chondrites varies from 2.1 to 3.5 g/cm3; the density of enstatite chondrites varies from 3.5 to 3.8 g/cm3. The range of achondrite density values is smaller: from 2.8 to 3.3 g/cm3. The density of stony–iron meteorites is significantly greater: from 4.2 to 4.8 g/cm3. Iron meteorites have the greatest density: from 7 to 8.12 g/cm3 (Britt and Consolmagno 2004).

The impact velocities for asteroids, that may collide with the Moon, range from 6 to 48 km/s (Yue et al. 2013), with mean value of 17.4 km/s.

Comets that can potentially collide with the Moon mostly belong to three types: Jupiter-family short-period comets, Saturn-, Uranus- and Neptune-family comets (including Halley’s comet), and long-period comets. Short-period comets are the ones whose solar orbital period does not exceed 200 years. Estimations (Artemieva and Shuvalov 2008; Chyba et al. 1994; Ong et al. 2007) found that the velocity of these comets is about 20–25 km/s. The solar orbital period of long-period comets exceeds 200 years, while the velocity of these comets is higher—about 55 km/s (Jeffers et al. 2001; Ong et al. 2007) or, according to different data, 70 km/s (Berezhnoy et al. 2003). Comet density ranges widely: from 0.4 g/cm3 (Churyumov–Gerasimenko comet) to 1.2 g/cm3 (Ong et al. 2007). Comet size varies from 1 to 20 km in diameter. At present, the probability of collision between short-period comets like Halley’s comet and the Moon is 9 × 10−3 per 1 million years (Berezhnoy et al. 2003). For long-period comets, this probability is 2 × 10−2 per 1 million years, according to data (Berezhnoy et al. 2003). A number of researches (Shevchenko 1999; Shevchenko et al. 2007) made assumptions regarding the existence of so-called “comet showers” in the past. The Oort cloud, located on the outskirts of the Solar system, 100,000 au from the Sun, was considered the source of comet showers (Emel’yanenko et al. 2007). Gravity disturbances, caused by the Sun drawing near other stars, increase the flux of comets in the Solar system. The frequency of collisions between comets and the Moon during such showers increases up to 1–2 collisions per 1 million years, according to estimations (Emel’yanenko et al. 2007). The research by Wetherill (1976) assumed that half the terrestrial impact craters could be caused by comet impacts.

2.2 Impact Crater Formation

The formation of an impact crater from a collision of a silicate impactor with the surface of a terrestrial planet was described in detail (Melosh 1989). The research divides the crater formation process into three stages: contact and compression stage, excavation stage, and modification stage. During the first stage, the impactor (meteorite, comet, etc.) collides with the surface of the target (planet, satellite). At the point of impact, the substance of the target compresses and accelerates to a velocity, comparable to that of the impactor. The shockwave spreads in the target and in the impactor. The shock pressure partly or fully melts and vapourises the impactor’s substance and partly melts and vapourises the target’s substance. After the compression and depression wave passes through the entire impactor, the impact stage is over. This stage is the shortest one. Its duration is less than 1 s and is determined by the impactor size, composition, and velocity, as well as by the composition of the target (Melosh 1989).

The excavation stage commences after the contact and compression stage. During this stage, a decompression wave emerges due to the rebound of the shock wave at the back end of the impactor. This stage forms a so-called “transient crater”, whose size is significantly greater than that of the impactor. In this stage part of the target material ejected from the future crater, and some—is displaced downwards and from the center of the shaft under the crater. The depth of the excavation is only about 1/3 of the depth of the transition of the crater. Material, which is discarded in the formation of a transitional crater, is moved along ballistic trajectories. Continuous emissions cover is formed at a distance of about one diameter of the crater wall. The greatest emission deposits in the crater have the power shaft. For continuous cover emissions abroad are arranged spots and have a small capacity. The duration of this stage, similarly to the first case, is determined by the parameters of the impactor and the target. On average, it lasts from several seconds to minutes (Melosh 1989).

The last stage of crater formation is the modification stage, which commences from the moment when the formed transient crater starts collapsing under the effect of gravity. This stage lasts throughout the existence of the crater. Crater walls crumble and slide, terraces and landslides emerge. Debris accumulates on the bottom, reducing the depth of the crater. In addition, the crater is exposed to the destructive effect of following meteorite impacts.

There are three main types of impact craters (Melosh 1989): simple and complex and multi-ring basins. Small bowl-shaped craters are regarded as simple craters. On the Moon their diameters do not exceed 10–18 km and their depths do not exceed 3 km (Melosh 1989). Multi-ring basins are found not only on the Moon (East Sea), but also in other objects of the Solar System (the Caloris plains on Mercury). The Moon in the peripheral ring structures appear, since the diameter of 140 km.

Complex craters have a flat bottom, terraces on their walls and a central elevation (mount or peak). The relative depth of complex craters are smaller than that of simple craters of the same diameter (Melosh 1989). The diameters at which transition starts from simple to complex craters depends on the gravity of the Moon and is in the range of 10–18 km.

2.3 Preservation of Impactor Substance in the Crater After Its Formation

It is widely accepted that during a crater formation, the impactor substance vapourises or scatters far beyond the formed crater. However, recent studies showed this was not always the case. A study by Melosh and Pierazzo (2000) simulated the falls of impactors at various angles and velocities on the lunar surface and studied the effect of impact velocity and angle on the amount of vapourised impactor substance. It was shown that the ratio of vapourised substance reduced with the impact angle—more than half the impactor substance was preserved after the end of the process at a 45° impact. At <30° impacts, most of the impactor substance is ejected from the forming crater during the early stages and can probably be found in ejecta deposits. At >60° impacts, most of the preserved impactor substance remains inside the crater.

Bland et al. (2008) showed that when an impactor collides with the Moon at <7 km/s and 15°–90°, most of the impactor would remain solid, but broken into fragments. According to obtained results, at 45°–90° impacts, more than 50 % of the impactor will remain in the formed crater. Furthermore, at <5 km/s velocities and near-vertical impact angles, the ratio of impactor substance that remains in the final crater can be 80 % and more. According to estimations (Bland et al. 2008), ~2.6 % lunar craters could contain >50 % of substance from the impactor that formed them. The authors (Bland et al. 2008) note that distinguishing craters, formed by a low-velocity impact (v < 7 km/s) is difficult, since while a high-velocity impact is expected to form a deeper transient crater with a greater wall slope, during the transformation of the transient crater into the final one it becomes more exposed to crumbling and terracing, hence becoming shallower, while the slope of its walls becomes gentler. Thus, it eventually becomes hardly distinguishable from a crater, formed by a low-velocity impact.

Bland et al.’s conclusions match the results, provided in the paper by Yue et al. (2013), which proved that when a lunar crater with a diameter of >20 km is formed by an impactor, whose velocity is higher than 14 km/s, most of a dunite impactor evaporates, but at an impact velocity v < 12 km/s, most of the impactor substance is preserved. At that, the impactor material ends up fragmented, perhaps partly melted, but most of it remains in the crater. In ordinary small craters, this substance is scattered among ejecta and deposited in breccia on the crater bottom. In complex craters, most of this substance is accumulated in the central peak during the collapse of the transient crater, at that, part of the substance remains in breccia that fills the bottom of the final crater, its ejecta and on the rim. According to estimations (Yue et al. 2013), about 25 % of impacts of asteroids and meteorites with the Moon occurred at velocities of <12 km/s. Thus, according to authors’ estimations, impactor substance could be found in about a quarter of lunar craters with a diameter of >20 km.

Potter and Collins (2013) investigated the effects of such factors as target material, impact angle and velocity, impactor porosity and shape on asteroid survivability. They found that at impact velocity of 5 km/s the whole dunite impactor remains solid, at an impact velocity of 12 km/s approximately 41 % of impactor mass remains solid, at impact velocities of 15 km/s and of 20 km/s approximately 10 and 3 % substances of impactor remains solid respectively.

The preservation of impactor substance at asteroid and cometary impacts has been investigated in Svetsov and Shuvalov (2015). According to their estimations at the impact angle of 45° almost 50 % of impactor mass (for ordinary chondrite) remains in the crater at impact velocity ≤12 km/s and almost 90 % at impact velocity ≤9 km/s. They also found that for carbonaceous chondrites at the impact angle of 45° approximately 30, 35 and 40 % of impactor mass remains solid at impact velocities 12, 9 and 6 km/s respectively.

A possible confirmation of impactor substance preservation on the Moon’s surface could be the finding of new rock types, which did not fit into the conventional model of the lunar crust evolution, by the Moon Mineralogy Mapper (M3) that operated aboard the Chandrayaan-1 probe in 2008–2009 (Pieters et al. 2010), and the spectrometer of the Japanese SELENE lunar orbiter (Yamamoto et al. 2010). These rocks were named OOS for the names of minerals they contained, such as orthopyroxene, olivine, and spinel. Olivine was discovered on the lunar surface in the middle of the last century (McCord et al. 1972), but a large amount of spinel was found for the first time. According to data (SELENE/KAGUYA 2010), olivine exposures are common on the outskirts of impact basins and are not found on the far side of the Moon in areas with a thick highland crust. This produced the assumption regarding the mantle origin of olivine. Nevertheless, the research of Sun and Li (2014) notes that half of olivine exposures are not related to large impact basins, while many olivine exposures demonstrate the presence of crystal plagioclase, which rules out their mantle origin, since plagioclase is a material from the top crust layers. Thus, it cannot be deposited in the peak at the same time as olivine, since olivine is not found in shallow crust layers. The research of Pieters et al. (2010) hypothesized that since main belt asteroids were the only other spinel-rich bodies, it was possible that discovered materials were the remnants of an asteroid that collided with the Moon.

Another important evidence in favor of the possibility of impactor substance preservation in the impact crater or its ejecta is the discovery of minerals that are part of chondrites (in particular, olivine and serpentine) on the surface of the Vesta asteroid by the Dawn probe that studied the asteroid in 2011–2012 (McCord et al. 2012; Nathues et al. 2014; Reddy et al. 2012). Areas with increased olivine concentration were identified in 15 impact craters on the asteroid. According to research data, all areas that contain substance of chondrite composition are located inside or near impact craters. The presence of olivine on Vesta was also recorded by ground surveillance and the Hubble space telescope (Le Corre et al. 2015).

It is worth noting that the data of the gamma ray spectrometer (GRS) and neutron spectrometer (NS) aboard the MESSENGER (MErcury Surface, Space ENvironment, GEochemistry and Ranging) probe showed that carbon concentration on Mercury’s surface was 4.1 % according to the GRS and 1–5 % according to the NS (Peplowski et al. 2015). This concentration is significantly higher than that of Earth, the Moon, and Mars. The research by Nittler et al. (2011) showed that the composition of Mercury’s surface corresponds with the partial silicate melting of enstatite chondrites. Enstatite chondrites have a relatively high carbon concentration (0.06–0.6 %), part of it in the form of graphite. Exogenous introduction of substance on Mercury’s surface is also related to the discovery of an increased (1.9 % in the northern hemisphere) concentration of Fe, compared with the predicted value (<0.6 %). However, Cook et al. (1997) showed that the excess of graphite may be consistent with geochemical models of Mercury’s differentiation.

2.4 Shackleton Crater Characteristics

Shackleton crater (89.9°S, 0.0°E) is 21 km in diameter and is located near the south pole of the Moon and the rim of the largest impact basin on the Moon—the South Pole–Aitken basin. This crater is special because the Lunar Orbiter Laser Altimeter (LOLA) aboard the Lunar Reconnaissance Orbiter (LRO) found its depth to be significantly greater than the depths of similar craters in terms of preservation and size [Shackleton is up to 4 km deep (Fig. 1), while the average depth of similar craters is 2.5 km].

Fig. 1
figure 1

Altitudes (in km) near Shackleton crater, according to data from LOLA onboard the probe LRO (EXPLORER 2015)

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With the LOLA data, the authors built a terrain and profile of Shackleton crater by the directions, indicated in Figs. 1 and 2.

Fig. 2
figure 2

Vertical profile of Shackleton crater. Directions indicated in Figs. 1 and 2 (a AB profile, b CD profile)

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These data were used to assess the morphometric parameters of Shackleton crater; the results are presented in Table 1. The volume of the inner cavity is 660 km3; the volume of the crater cavity below the surface level is 316 km3; the volume of the rim is 390 km3. The crater is up to 4 km deep, which is significantly deeper than craters of similar size and degree of preservation, which are, according to data (Rodionova and Dekhtyareva 1988), 2–3 km deep. The “depth–diameter” ratio of Shackleton is 0.2, while the average value of this ratio for other similar lunar craters is 0.13. The crater has a flat bottom, whose average diameter is 6.8 km. The rim diameters in the AB and CD directions have a similar ridge and differ at the level of the surrounding surface (Table 1). The crater rim is symmetrical, but its inner cavity is elongated, with a greater outer slope in the CD direction. The deepest point of the crater does not coincide with its center and is shifted along the CD profile at 2.2 km (Fig. 2b). It reaches the depth of −2.81 km. In the paper by Bottke et al. (2000), crater ellipticity ε was estimated as the ratio of the largest and smallest diameter. The crater was considered elliptic if ε ≥ 1.1. In Shackleton’s case, this value is 1. Thus, according to Bottke et al. (2000), Shackleton is not an elliptic crater. At the same time, Shackleton has different crater diameters at the level of the surrounding surface: the crater is 0.8 km longer in the CD direction (Table 1). Davison et al. (2011) defines crater ellipticity as the ratio of the largest and smallest diameter at the surface level and introduces several other parameters, including offset O. Offset O is equal to the ratio of the distance to the deepest point of crater from the wall to the length of crater, their length is the maximal crater diameter. For Shackleton the maximal crater diameter is diameter alone profile CD and it is equal 17.1 km (Table 1). Thus, offset is equal 0.488. Denote ellipticity according to Davison et al. (2011) as ε1, to differentiate it from ε according to Bottke et al. (2000). In the research of Davison et al. (2011), similar to that of Bottke et al. (2000), the ellipticity criterion is ε1 ≥ 1.1. In Shackleton’s case, ε1 = 1.049, while O = 0.488 according to Davison et al. (2011) 40°–50° angle.

Table 1 Morphologic parameters of Shackleton crater
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Previously it was thought that the Shackleton crater was the same age as the Copernican period (<1 billion years) (Lavoie 2006). However, the work by Spudis et al. (2008) estimated the crater density by the deposits of Shackleton’s ejecta and showed that it could date to the Imbrian period, while its age could be up to 3.6 billion years. Figure 3 shows the geological map of the south pole of the Moon from the research by Spudis et al. (2008). According to this map, Shackleton crater material are arranged symmetrically with regards to its center; they spread for up to 8.1 km from the rim. As seen in Fig. 3, Shackleton’s crater material are undisrupted throughout their entire extent, because they are not covered by other younger deposits. In the 180° E direction, these crater material cover older deposits that date back to the Nectarian period (3.9–3.8 billion years); in the 0° E and 90° E direction, the crater material cover pre-Nectarian deposits (4.5–3.9 billion years).

Fig. 3
figure 3

Geological map of the lunar South Pole (Spudis et al. 2008). Crater Shackleton lies near the South Pole. The position of the crater indicated by arrow. Surface units marked as: c—crater material, Ip—plains material, m—massif material, pl—platform massif material, sc—satellitic (basin secondary) crater material (Spudis et al. 2008). Different colors indicate different geological periods: Copernican—yellow, Eratosthenian—green, Imbrian—blue, Nectarian (N) and pre-Nectarian (pN)—brown. The boundary of the Shackleton crater material is marked by a white line

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It is expedient to estimate the parameters of the impactor (such as its density, size, velocity, and impact angle), which could have formed the Shackleton crater. To do so, it is necessary to estimate the parameters of the transient crater, which was formed at the end of the excavation stage, after the ejection of the target’s rocks. According to Croft (1985), the transient crater diameter is determined by the following equations:

$$ D = 1.3D_{t} \quad if\;D_{t} \le {{D_{0} } \mathord{\left/ {\vphantom {{D_{0} } {1.3,\;and\;D = 1.4\left( {{{D_{t}^{1.18} } \mathord{\left/ {\vphantom {{D_{t}^{1.18} } {D_{0}^{0.18} }}} \right. \kern-0pt} {D_{0}^{0.18} }}} \right)}}} \right. \kern-0pt} {1.3,\;and\;D = 1.4\left( {{{D_{t}^{1.18} } \mathord{\left/ {\vphantom {{D_{t}^{1.18} } {D_{0}^{0.18} }}} \right. \kern-0pt} {D_{0}^{0.18} }}} \right)}}\quad if\;D_{t} \ge {{D_{0} } \mathord{\left/ {\vphantom {{D_{0} } {1.3,}}} \right. \kern-0pt} {1.3,}} $$
(1)

where D is the final crater diameter, Dt is the transient crater diameter, D0 is the crater diameter, at which simple craters become complex ones for a given body. As shown in the research by Pike (1977), in the Moon’s case, this value ranges from 15 to 20 km. Using the formula (1) we obtain that the transient crater diameter for crater with real diameter 21 km is 11 km < Dt < 16.2 km. Since the transient crater depth Ht, as shown in Melosh (1989), constitutes from about 1/4 to 1/3 of the transient crater diameter, for Shackleton this value will range from 2.8 km ≤ Ht ≤ 5.3 km, with the supremum of Ht being smaller than the real depth of Shackleton crater, which, according to LOLA data, is 4 km. By replacing the Ht supremum with the observed depth of Shackleton crater, one obtains the new range for the transient crater diameter: 15.6. km < Dt < 16.2 km. The volume of the transient crater (Vt) is estimated on the assumption that it has a sphere segment shape:

$$ V_{t} = {1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-0pt} 6}\pi H_{t} \left( {H_{t}^{2} + 3R_{t}^{2} } \right), $$
(2)

where Rt is transient crater radius. According to our calculations the transient crater volume is 386 km3 ≤ Vt ≤ 625 km3. The diameter and volume of the transient crater depend of many parameters, including impactor density and size, density of the target’s substance, impact velocity and angle. As demonstrated by works (Holsapple and Housen 2007; Schmidt and Housen 1987) in gravity drive, if the target’s rocks are not porous, these dependencies can be expressed by the following equations:

$$ D_{t} = 1.161\left( {{{\rho_{i} } \mathord{\left/ {\vphantom {{\rho_{i} } {\rho_{t} }}} \right. \kern-0pt} {\rho_{t} }}} \right)^{1/3} D_{i}^{0.78} v_{i}^{0.44} g^{ - 0.22} , $$
(3)
$$ V_{t} = 0.28\left( {{{\rho_{i} } \mathord{\left/ {\vphantom {{\rho_{i} } {\rho_{t} }}} \right. \kern-0pt} {\rho_{t} }}} \right)D_{i}^{2.35} v_{i}^{1.3} g^{ - 0.65} , $$
(4)

where ρi is the impactor density, ρt is the target’s material density, vi is the impact velocity, Di is the impactor size, and g is the gravitational acceleration (for the Moon, it is 1.622 m/s2). Formula (4) concerns a vertical impact. In case of an angled impact, vi should be replaced with vi sin Θ, where Θ is the impact angle, measured off the target surface plane.

Since Shackleton crater is located in the highland area with no traces of basaltic lava in its vicinity (Spudis et al. 2008), assume the density of the lunar surface in this area is 2.7 g/cm3, which matches the anorthosite density. The abovementioned densities of meteorites and comets will be considered the impactor density. It is necessary to estimate the assumed impact angle. The research by Melosh (1989) noted that the impact angle primarily affects the ejecta form, rather than the crater itself. For example, the author showed that ejecta changed form already at a ≤60° impact angle, while the crater rim remained round even if the impact angle reduced to 10°. In the research of Herrick and Fosberg-Taylor (2003), the authors studied the dependence of crater form on the impact angle. It was shown that crater ejecta on the Moon became asymmetrical at ≤45° impact angles. In addition, the authors showed that at ≤15° impact angles, the crater rim became saddle-shaped, while at ≤10° impact angles, the crater rim was elongated in the impact direction. Shackleton crater material deposits, located around its rim, appear to be symmetrical; thus, it is possible to assume that the impact angle was ≥45°. However, the observed asymmetrical shape of Shackleton crater at the level of the surrounding surface, according to Davison et al. (2011), means that the impact angle ranges from 40° to 50°. Therefore, it is possible to assume that the impact angle that formed the Shackleton crater is 45° ≤ θ ≤ 50°.

We used formulas (3) and (4) to calculate the transient crater diameter (Dt) and volume (Vt) for different types of impactors. The impactor characteristics, used in the present research, are presented in Table 2. The minimum value of comets velocity of 3 km/s has been taken from the work of Bland et al. (2008).

Table 2 Types and characteristics of impactors that collide with the Moon
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We compared the calculated values of transition crater diameter and its volume with values of these parameters estimated on the basis of the morphological characteristics of Shackleton crater. As a result of potential 283 impactors had been received. Modeling results presented in Figs. 4 and 5 shows the distribution of possible impactors by such characteristics as size and velocity. The upper limit of the size of impactors (20 km) was chosen almost equal to the diameter of the crater considering, that the crater size cannot be smaller than impactors, which have formed it. The lower limit of the impactor sizes, we have chosen equal to 0.5 km. Ivanov (2008) showed that moon crater with of 20 km can be formed by the falling asteroids with diameters of 1 km.

Fig. 4
figure 4

Distribution of possible impactors by sizes

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Fig. 5
figure 5figure 5

Distribution of possible impactors by velocities (a comets, b chondrites, c achondrites, d stony–iron, e iron–nickel), where 100 % is the total number of possible candidates—283

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The number of impactors is given in %, where 100 % is the total number of possible candidates—279. Characteristics of potential impactors shown in Table 3. According to the our estimations, most impactors were comets (42.3 %); the percentages of chondrites and achondrites were 22.6 and 19 %, respectively. The percentage of stony–iron impactors is about 4 % of the total number of possible candidates. The percentage of iron–nickel impacts is 12 %. Figure 4 shows that the size of possible chondrites and achondrites does not exceed 2 km. The maximum size of stony–iron and iron–nickel impactors—2 and 1.5 km respectively. The maximum size of comets reaches 6 km. Comets with a size of 0.5 km are not encountered among possible candidates.

Table 3 Characteristics of potential impactors
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As seen in Fig. 5a, comets are divided into two groups according to velocity: short-period comets with velocity ranges from 3 to 48 km/s, and long-period comets with velocity ranges from 58 to 68 km/s.

The range of chondrite velocity values is from 6 to 25 km/s. The distribution has three maxima, at 6, 8 and 17 km/s (Fig. 5b).

The velocities of potential achondrites impactors lie in two ranges: 6–9 and 16–20 km/s (Fig. 5c). Stony–iron impactors are presented in the velocities intervals of 6–7 and 12–15 km/s (Fig. 5d). Iron–nickel impactors are also separated into three groups: 6, 9–10 and 28–36 km/s (Fig. 5e).

The dependency between the volume and velocity of ejecta during the formation of an impact crater and the volume and velocity of the impactor was studied in the paper by Bland et al. (2008). In particular, the research showed that the ratio between the ejecta volume and velocities, exceeding a certain ve value, the impactor volume and impact velocity could be expressed with the following equation:

$$ {{V\left( {v_{e} } \right)} \mathord{\left/ {\vphantom {{V\left( {v_{e} } \right)} {V_{pr} = 0.01385\left( {{{v_{i} } \mathord{\left/ {\vphantom {{v_{i} } {v_{e} }}} \right. \kern-0pt} {v_{e} }}} \right)^{1.66} }}} \right. \kern-0pt} {V_{pr} = 0.01385\left( {{{v_{i} } \mathord{\left/ {\vphantom {{v_{i} } {v_{e} }}} \right. \kern-0pt} {v_{e} }}} \right)^{1.66} }} $$
(5)

where Vpr is the impactor volume, vi is the impact velocity and ve is the ejecta velocity, V(ve) is the ejecta volume with velocity, exceeding ve.

We used formula (5) to calculate the volume of impactor, which could form the Shackleton crater. To do this, we estimated the ejecta volume for Shackleton crater, located near its rim and identified as its deposits according to Spudis et al. (2008). As shown in the work by McGetchin et al. (1973), the thickness of ejecta deposits over the crater rim (T), the dependency of deposit thickness on the distance from the transient crater rim (r), and volume of ejecta beyond distance r from the crater rim (V) and the total volume of ejecta (VT) are determined by the following formulas:

$$ T = 0.14R_{t}^{0.74} $$
(6)
$$ t = T\left( {{r \mathord{\left/ {\vphantom {r {R_{t} }}} \right. \kern-0pt} {R_{t} }}} \right)^{ - 3} $$
(7)
$$ V_{T} = 2\pi TR_{t}^{2} $$
(8)
$$ V = V_{t} {R \mathord{\left/ {\vphantom {R r}} \right. \kern-0pt} r} $$
(9)

As mentioned previously, for Shackleton crater 15.6 km < Dt < 16.2 km and transient crater radius is 7.8 km < Rt < 8.1 km respectively. Therefore, according to formula (6), the thickness of ejecta deposits over the transient crater rim for Shackleton crater will be about 0.106 < T < 0.109 km, while the total volume of ejecta from the crater is 41 km3 < VT < 44.5 km3. At the same time, the work by Pike (1974) showed that the thickness of ejecta deposits over the transient crater rim could be expressed by a different equation: T = 0.033Rt. In this case, the thickness of Shackleton’s ejecta deposits over the rim is significantly greater—0.258 km < T < 0.266 km, while the total volume of ejecta is 99 km3 < VT < 109 km3. According to the data, provided in the research of Petro and Pieters (2006), T = 0.0078RT. At that, the total volume of ejecta will be 24 km3 < VT < 26 km3. As shown previously, according to Fig. 3, the boundary of undisrupted crater material is located 8.1 km away from the rim, on average. Formula (9) shows that the volume of ejecta at this distance is V = 51 km3 according to McGetchin et al. (1973) and V = 81 km3 according to Pike (1974).

It is necessary to estimate the volume and velocity of ejecta beyond the 8.1-km-distance from the Shackleton crater rim. The volume of such ejecta amounts to 16 km3 < VT < 19 km3 according to McGetchin et al. (1973), to 38.7 km3 < VT < 47 km3 according to Pike (1974), and to 9 km3 < VT < 11 km3 according to Petro and Pieters (2006). It is expedient to estimate the ejecta velocity with the formula from the research of Melosh (1989), which expresses the dependency of the ballistic range Rb on velocity:

$$ R_{b} = \left( {{{v_{e}^{2} } \mathord{\left/ {\vphantom {{v_{e}^{2} } g}} \right. \kern-0pt} g}} \right)\sin \left( {2\phi } \right) $$
(10)

where φ is the ejection angle. The ejection angle can vary widely, but most ejections, according to Hermalyn and Schultz (2010), occur at angles of 40°–45°. In this case, the ejection velocity for deposits, located 8.1 km away from the crater rim, amounts to 0.11 km/s. With the obtained values of ejection velocity and volume, according to the data of McGetchin et al. (1973), Petro and Pieters (2006), Pike (1974), and the data from Table 2, formula (5) can be used to estimate the characteristics of possible impactors, whose impact could have formed the Shackleton crater. The results are presented in Table 4.

Table 4 Characteristics of possible impactors, whose impact could have formed the Shackleton
Full size table

We obtained that the possible impactors that formed the Shackleton crater are low-velocity comets (3 km/s) and asteroids (velocity ≤ 12 km/s), with a diameter of 4–4.5 km for comets and 1–2 km for asteroid, which collided with the lunar surface at a 45°–50° angle.

The estimation of the volume and thickness of ejecta by the Petro and Pieters (2006) and McGetchin et al. (1973) models did not include chondrites and achondrites as possible impactors. According to Petro and Pieters (2006) model the possible impactors that formed the Shackleton crater are comets, stony–iron and iron–nickel impactors. Stony–iron impactors and comets are encountered among possible candidates only when using the McGetchin et al. (1973) model of thickness of ejecta deposits. The size of iron–nickel impactors is 1–2 km. The size of stony–iron impactors ranges from 1 to 1.5 km. Chondrites and achondrites impactors are encountered among possible candidates only when using the Pike (1974) model. The size of all possible chondrite and achondrite impactors is 2 km (Table 4).The velocity of chondrite and achondrite impactors is 6 km/s. The velocity of iron–nickel impactors ranges from 6 to 9 km/s. Only one stony–iron impactor have 12 km/s.

We calculated the impactor mass inside the crater Shackleton using models from Bland et al. (2008) for all types of impactors, excluding comets, and model from Svetsov and Shuvalov (2015) for ordinary and carbonaceous chondrites and comets (Table 4). We obtained that according to conclusions in the paper by Bland et al. (2008), since the present study estimated the possible impact angle at ≥45° and the velocity at ≤6 km/s, more than 50 % of the impactor mass could be present inside the crater. Thus, from 6.3 × 1012 to 7.3 × 1012 kg of chondrite or achondrite asteroid mass, or from 1.8 × 1012 to 7.1 × 1012 kg of iron–nickel asteroid substance, or from 1.2 × 1012 to 8.9 × 1012 kg of stony–iron asteroid substance could have remained inside the Shackleton crater (Table 4). According to Svetsov and Shuvalov (2015) from 1.2 × 1013 to 1.3 × 1013 kg of ordinary chondrites asteroids mass and from 5.2 × 1012 to 5.8 × 1012 kg of carbonaceous chondrites can remain in the crater Shackleton.

Svetsov and Shuvalov (2015) estimated the fraction of comets which could remain in a crater after impact. They found that at velocities 8–10 km/s only less than 1 % of comets substances remain in the crater. We used this value for the estimation of lower mass limit of the comet material which may remain in the crater after the impact. We obtained that no less than 6.3 × 1012 kg comet substances can remain in the crater Shackleton (Table 3).

The amount of impactor which remains solid, melted or vaporized after impact of chondrite impactors was estimated using model from work Potter and Collins (2013). To do this we calculated the fraction of impactor that experiences a peak shock pressure less than P according to Potter and Collins (2013), who found that at shock pressure <50 GPa impactor is weakly shocked and remaining solid at shock pressure <105 GPa. The incipient vaporization pressure for impactor is 186 GPa. Impactors parameters were taken from Table 3. In our calculations we approximated the target as granite and impactors as dunite. We obtained from 96 to 99 % of chondrite impactor remain solid and from 1 to 4 % of the mass of the impactor melted after impact.

3 Results and Discussion

The conclusion that the Shackleton crater could have been formed by such impactors as stony–iron and iron–nickel asteroids is confirmed by data from the research of O’Keefe and Ahrens (1982), which showed that with identical other conditions, the smaller the impactor density, the smaller the “depth–diameter” ratio. The greater value of this ratio in case of the Shackleton crater, compared with most other lunar craters with a similar diameter and degree of preservation, assumes that the crater was formed by a dense impactor, perhaps even of stony–iron or iron–nickel composition. Obtained small values of impact velocity are in line with the results of the research by Le Feuvre and Wieczorek (2008), which studied the dependency of the velocity of impact with planetary surfaces and the lunar surface. The paper showed that the impact velocity in the polar areas of the Moon was 0.83 times smaller than the impact velocity in its equatorial areas.

4 Conclusions

The present paper attempted to estimate the parameters, such as velocity, impact angle, size and density, of the impactor that formed the Shackleton crater on the Moon. The authors used some peculiarities of the Shackleton crater, such as its greater depth (4 km), compared with other craters of similar sizes, and certain asymmetry of the crater at the level of the pre-impact target surface, which narrowed down the range of transient crater parameters and allowed estimating the impact angle. The authors concluded that the morphometric parameters of the crater could indicate that the impact occurred at a 45°–50° angle, while the transient crater depth ranged from 4 to 5.3 km. Possible impactors included short-period comets, with a size of 4–4.5 km and velocity of 3 km/s and asteroids of a chondritic, achondritic, stony–iron, and iron–nickel composition, with a size of 1–2 km and velocity of 6–12 km/s. Obtained values of impact angle and velocity allow assuming that the crater and its deposits could contain part of the impactor substance more than 50 % for asteroids according to estimations (Bland et al. 2008) and about 1 % for comets according to Svetsov and Shuvalov (2015). The main part of the impactor material (>90 %) can be in the solid state.

Shackleton crater is not currently regarded as one of the priority landing sites for future lunar missions (Lemelin et al. 2014). This is due to the fact that, despite the presence of the crater permanent shaded areas (Bussey et al. 2010), the maximum temperature in the crater Shackleton too high [from 81 to 105 K (Lemelin et al. 2014)] for the existence of the deposition of any volatile compounds except for H2O.

However, Shackleton crater might be of interest as a source of an asteroid or comet substance that may remain in the crater and beyond around the shaft after the impact. Different minerals, and volatile compounds might remain in the asteroid and comet substance. They can serve as a source of titanium, nickel, iron and other elements (Shevchenko 2014). The asteroid Psiheya which contains 1.7 × 1019 kg iron–nickel ore (in 100 thousand times higher than that of ore reserves in the Earth’s crust) is a good example.