Lunar regolith and water ice escape due to micrometeorite bombardment

Journal Pre-proof Lunar regolith and water ice escape due to micrometeorite bombardment

J.P. Pabari, S. Nambiar, V. Shah, A. Bhardwaj PII:

S0019-1035(18)30679-1

DOI:

https://doi.org/10.1016/j.icarus.2019.113510

Reference:

YICAR 113510

To appear in: Received date:

16 October 2018

Revised date:

26 September 2019

Accepted date:

26 October 2019

Please cite this article as: J.P. Pabari, S. Nambiar, V. Shah, et al., Lunar regolith and water ice escape due to micrometeorite bombardment, (2019), https://doi.org/10.1016/ j.icarus.2019.113510

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© 2019 Published by Elsevier.

Journal Pre-proof Lunar Regolith and Water Ice Escape due to Micrometeorite Bombardment J. P. Pabari1*, S. Nambiar1, V. Shah2 and A. Bhardwaj1 1

PRL, Ahmedabad, INDIA, 2

CSPIT, Changa, INDIA

*E-mail: [email protected]

Abstract

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Dust particles exist everywhere in interplanetary space and they evolve dynamically after their origination from the sources like Asteroid belt, Kuiper belt, comets or space debris left during

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the formation of solar system. These micrometeorites encounter the inner planets, while they spiral-in towards the Sun. From whichever come to Earth, many particles are ablated in the Earth’s atmosphere

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and leave the metallic ions behind. In case of Moon, all such particles can reach the surface without

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ablation owing to the absence of atmosphere. Due to the impact of hypervelocity dust particles on lunar surface, ejecta come out in the lunar environment. In some cases, the ejecta velocity could be

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larger than the escape velocity and particles may be able to escape from Moon. Further, the escaping ejecta may carry water ice (volatiles), whenever incoming projectiles hit the surface in polar region with the water ice present. In this paper, we have computed the ejecta parameters and estimated the

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possible escape of volatiles from Moon, using Galileo observations of the dust particles near Moon. Considering the incident angle distribution, the upper limit of regolith escape rate is found to be

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~2.218 × 10-4 [1.662 × 10-4, 10.232 × 10-4 ] kg/s. Similarly, the upper limit of water ice escape rate is found to be ~1.988 × 10-7 [1.562 × 10-7, 7.567 × 10-7 ] kg/s. On one side, Moon is found to be gradually becoming heavier due to its one order higher incoming dust particles than those escaping from it. While on the other side, Moon could be depleted of water ice (volatiles) resources over a period of time, because of the escape due to micrometeorite impact. The results presented here could be useful to understand the dust and volatile escape from Moon.

Key Words: Escape, Micrometeorite, Moon, Regolith, Water Ice

1. Introduction and Motivation Dust particles are found everywhere in the interplanetary space in our solar system and they evolve dynamically after their origination from sources like Kuiper belt, Asteroid belt, space debris left during the formation of solar system or comets. Such dust particles 1

Journal Pre-proof spiral-in towards the Sun due to its gravitational pull and they encounter a planetary body on their ways. There is a continuous shower of micrometeorites in planetary system and Earth, being a member of it, is not the exception. It is known that every year (40 ± 20)  106 kilograms (Love and Brownlee, 1993) of dust is accumulated on Earth from outside the planet. Because of the continuous bombardment of micrometeorites on Earth and also its natural satellite, i.e., Moon, ejecta come out from the surface and leave dust particles in the environment. From whichever come to Earth, many particles are ablated in the Earth’s atmosphere and they leave metallic ions behind. Those particles, which can survive in the atmosphere, reach the surface. It is difficult to explain the escape of ejecta from Earth,

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because of its larger escape velocity (~11 km/s). However in case of Moon, all incoming

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particles can reach the surface without ablation owing to the absence of atmosphere. The impact of hypervelocity dust particles on lunar surface causes the ejecta to come out in the

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lunar environment. Particles ejected due to impact of a hypervelocity particle can have maximum velocities in the range of incoming impact velocity. The velocity of each ejected

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particle depends on its mass as well, along with the incoming particle velocity and mass.

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Since the incoming particles have velocity in the range of ~10-50 km/s, a portion of the ejected particles will always have velocity greater than 2.39 km/s (the escape velocity of Moon). Thus, some of the particles with velocities larger than the escape velocity of Moon

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are able to escape from it. For instance, for a dust particle travelling at a speed of 18 km/s (Molina-Ciberos et al., 2003), there is a possibility that a part of the ejecta could have velocity

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larger than the escape velocity (2.39 km/s) of Moon, as shown in later part of the article. Soter (1971) has discussed the formation of dust belt around Mars, due to hypervelocity impact of dust particles on Phobos and Deimos. Schneider (1975) has given the estimation of secondary micro-crater density formed by the fast moving ejecta, which are generated from the incoming particles. Davis et al. (1981) have explained the ejecta escaping from natural satellites of Mars. Juhasz et al. (1993) have reported a dust halo around Mars, caused by the escaping dust particles from Phobos and Deimos. Ishimoto (1996) has discussed the possibility of formation of Phobos/Deimos dust rings. Krivov and Hamilton (1997) have shown the possibility of a dust belt existing around Mars, based on the particles escaping from its natural satellites. Francesconi et al. (2013) have reported various ejecta models and also, the experimental activities related to hypervelocity impact on spacecraft outer surface. Pabari and Bhalodi (2017) have compared the flux of particles escaping from Phobos/Deimos

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Journal Pre-proof (and reaching Mars) with the original flux of interplanetary dust particles directly reaching Mars, to understand the contributions of both at Mars. Further, it is known that surface of airless, non-magnetized Moon is directly exposed to the solar wind plasma and ultraviolet (UV) radiation. This causes the lunar surface to be electrically charged. The lunar surface acquires electrostatic potential while being exposed to UV from Sun on the day side, or due to plasma electron and ion currents on the night side. Significant temporal and spatial variations of the lunar surface potential are known to occur due to charging from photoemission and plasma currents and it can range from nearly +10 V to less than –500 V (Halekas et al., 2005, 2007, Manka, 1973). These forces due to electric

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fields can exceed gravity and surface forces (cohesion) for the small dust particles, causing

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them (Horanyi, 1996) to be levitated. Recently, Pabari and Banerjee (2016) have reported lunar surface charging and also discussed a possibility of dust levitation on Moon. In normal

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solar conditions, the levitated particles could remain bound within the gravity of Moon, while in case of extreme solar conditions like SEP, the surface voltages could be larger. Near the

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terminator, the sharp electric field gradient can cause larger velocities for the levitated dust

water ice molecules) is negligible.

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particles. However, such probability is less and the escape of levitated particles (bound to

Thus, major loss mechanism is governed by the bombardment of micrometeorites on

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lunar surface. Majority of incoming dust particles are Interplanetary Dust Particles (IDPs) in nature, possibly coming from sources like Asteroid belt and any space debris left behind at

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the time of solar system formation. Though comets leave particles whenever they travel near Sun, its possibility is lesser and it is not considered in our work. Canup et al. (2015) have reported that volatile depletion on Moon can be explained by preferential accretion of volatile-rich melt in the inner disk to Earth, rather than to the growing Moon. Desch and Taylor (2011) have described a model of Moon’s volatile depletion. Basically, volatile escape from Moon’s inner part (or rock) is found in literature. During Apollo time, a Lunar Ejecta And Micrometeorites (LEAM) experiment was performed on lunar surface to study micrometeorite impact and ejecta in the lunar environment. Li et al. (2015) have modelled impact generated ejecta from the lunar surface. Horanyi et al. (2015) have given LADEE orbiter based estimation of IDP flux reaching Moon and also the density of ejecta coming out due to dust impact. Since Moon does not have atmosphere, the micrometeoroid particles directly reach lunar surface. There are several studies and modelling results available in literature, characterizing the lunar surface and dust particle distribution near the surface. Gault et al. 3

Journal Pre-proof (1963) had modelled the dynamics of spray ejected from micrometeoroid impact on lunar surface. Such spray of micron and sub-micron sized particles characterize the fine lunar regolith. The particles ejected from surface contribute to increase the flux in vicinity of Moon. Morrison and Clanton (1979) have studied the size distribution of lunar micro-craters, caused by the impact of dust particles on lunar surface. Stubbs et al., (2006) as well as Pabari and Banerjee (2016) have studied levitated, charged dust particles, which may remain near the surface. Horanyi et al. (2015) have reported observations of a permanent, asymmetric dust cloud around Moon, caused by impacts of high-speed particles, using the Lunar Dust Experiment (LDEX) on LADEE mission. An empirical relation of the ejecta evolution due to

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impact of hypervelocity particles on several targets is available in the literature like Gault et

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al. (1963), Rival and Mandeville (1999) and Juhasz et al. (1993). Studies regarding volatile escape because of solar radiation, solar wind and gravitational escape are also available in the

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literature. Farrell et al. (2015) studied the spillage of volatiles (including water ice) to crater surroundings, because of solar wind and micrometeoroid impact. They have modelled the

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escape rate in terms of a number of molecules ejected from the floor of a Permanently Shadowed Region (PSR) crater. However, the escape of particles from lunar gravity has not

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been modelled. It is interesting for us to know the escaping dust ejecta and also to estimate the escaping water ice (volatile) from Moon. The escape process is important to understand

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the overall loss (in terms of dust and water ice, both) from the lunar surface. The water ice, being a potential source of energy, is important for future manned mission. All the earlier

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models like Gault et al. (1963), Rival and Mandeville (1999) and Juhasz et al. (1993), which are referred in this work, are based on the initial theory developed by Gault et al. (1963). Essentially, we have used this theory and selected the parameter values suitable for the Earth’s Moon. In the present work, we aim to model and quantify the flux of regolith as well as water ice (volatile) escaping from the lunar surface due to hypervelocity impact of dust particles. We have considered the particles with wide range of sizes and velocities for this purpose. We here report that some volatile could escape from the lunar surface due to dust particles, bombarding in the polar region. The escaping regolith and water ice (volatile) will be lost permanently to the space. Such lost particles could contribute as a source for the zodiacal dust cloud. Moreover, those particles, which cannot escape from the lunar surface, are lifted above the surface and contribute to the lunar environment. The present work can help provide the predictions of surface volatile escape from Moon over geological time, which may be useful to understand depletion of water ice in future.

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Journal Pre-proof The rest of the paper is organized as follows. Section 2 presents observation based, incident micrometeorites on Moon. Section 3 describes the impact theory and ejecta characteristics, Section 4 provides various results and investigates possible dust escape from Moon. Section 5 gives the estimation of water ice (volatile) escape from PSRs of Moon and the paper ends with conclusion.

2. Incoming Micrometeorites to Moon It is known that every planet in our solar system is continuously bombarded by micrometeorites. A big object hitting any planetary body may cause a bigger event, while

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smaller grains cause continuous erosion of the body. Grun et al. (1985) have given an

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estimate of incoming flux, derived from the measurements of Pegasus, Pioneer, HEOS and meteor observations. The Pegasus observations are reported by Naumann (1966) as the flux

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of 7.6 × 10-8 particle/m2 sec for particles with mass of 5.8 × 10-7 g and the flux of 3.1 × 10-7 particle/m2 sec for particles with mass of 9.3 × 10-8 g. Berg and Grun (1973) have reported

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the flux of 5.8 × 10-4 particle/m2 sec for the particle mass < 10-13 g, based on Pioneer 8 and 9 observations. Hoffman et al. (1975a, b) have reported the flux of 4.5 × 10-5 particle/m2 sec for

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the particle mass of 10-12 g, while Grun and Zook (1980) have provided the flux of 7.0 × 10-5 particle/m2 sec for the particle mass of 10-13 g; based on HEOS observations. Further,

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Whipple (1967) reported the flux of ~10-9 particle/m2 sec and the flux of ~10-17 particle/m2 sec, varying linearly between the particle mass of 10-4 g and 102 g. Thus, Moon experiences a

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continuous flow of incoming dust particles travelling at hypervelocity (> 1 km/s). A constant loss rate from Moon is assumed in our computation, which is based on Grun et al. (1985) model. The cumulative flux (particle/m2 s) of interplanetary particles in the mass range of 1018

g < m < 102 g is described by the Interplanetary Meteoroid Flux (IMF) model of Grun et al.

(1985), independent of time, for finding the flux of a given mass (or size), 𝐹(𝑚, 𝑟0 ) = (𝑐1 𝑚𝛼1 + 𝑐2 )𝛼2 + 𝑐3 (𝑚 + 𝑐4 𝑚𝛼3 + 𝑐5 𝑚𝛼4 )𝛼5 + 𝑐6 (𝑚 + 𝑐7 𝑚𝛼6 )𝛼7

(1)

where 𝑐1 = 2.2 × 103 , 𝑐2 = 15, 𝑐3 = 1.3 × 10−9 , 𝑐4 = 1011 , 𝑐5 = 1027 , 𝑐6 = 1.3 × 10−16, 𝑐7 = 106 in cgs units; 𝛼1 = 0.306, 𝛼2 = −4.38, 𝛼3 = 2, 𝛼4 = 4, 𝛼5 = −0.36, 𝛼6 = 2, 𝛼7 = −0.85 at 𝑟0 = 1 𝐴𝑈 and 𝑚 is the mass of the meteoroid in grams. Since, Moon is a natural satellite, orbiting around Earth, we take the distance from Sun to Moon as 1 AU. The change in the incoming flux of less than 10 % is expected by Banaszkiewicz and Ip

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Journal Pre-proof (1991), due to gravitational forces, shadowing of Moon by planet and orbital motion of Earth and Moon. The flux of particles reaching Moon is found from Eq. (1) and it is shown in Figure 1. From the results in Figure 1, one can find the accretion on Moon due to

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micrometeorite flux, as a corollary.

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Figure 1: Micrometeorite flux reaching Moon

For the computation in this paper, we need projectile mass (𝑚𝑖 ) and its velocity (𝜈𝑖 ). We found some literature, which considered single value of particle velocity. However, it is

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known that smaller particles travel with larger velocities and the larger particles travel with smaller velocities. Therefore, we have used mass and velocity data of the interplanetary

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particles, measured by Galileo Dust Detection System (DDS) near Moon. Galileo, after leaving Earth, had several flybys around different planetary bodies before reaching its destined orbit around Jupiter. We know that the average distance from Earth to Moon is 0.0025 AU. In our computation for impact generated ejecta on Moon, we have used mass and velocity of IDPs measured up to 0.1 AU distance from Earth. These closer particles are more likely to represent the parameters of IDPs reaching Moon (though, farther particles may also reach Moon). From the Galileo data given by Krueger et al. (2010), only valid data (decided by velocity error factor) have been used and a plot of logarithmic velocity and logarithmic mass is drawn, as shown in Figure 2. As depicted in Figure 2, a function is fitted to this plot and it is given by log 𝑣 = −0.18 log 𝑚 + 0.31

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(2)

Journal Pre-proof The mass-velocity function given in Eq. (2) is followed by the IDPs around Earth and therefore, also around Moon. Essentially, this indicates the fact that smaller particles travel faster and bigger particles travel slower. In the present work, further computation on impact generated ejecta is based on Eq. (2). This fitting exercise helps reducing the number of

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parameters in the impact theory used in this paper.

Figure 2: Measured velocity vs. mass given by Galileo (Krueger et al., 2010), up to 0.1 AU around Earth, which

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our computation.

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essentially represents the parameters of dust particles reaching Moon. A linear fit is also shown and it is used in

3. Micrometeorite Impact and Ejecta Characteristics Whenever an incoming micrometeorite falls on the lunar surface with large velocity (usually > 1 km/s as hypervelocity particle), it makes a pit on the surface as shown in Figure 3. The incoming particle has its own mass and velocity, giving energy to the system from its momentum. The impact creates the ejecta, which is made from fine grain regolith (and also volatile/water-ice in the permanently shadowed region) of the Moon. The ejecta may contain bunch of particles in a conical shape zone, the jets as well as the spalls. The jets are streams of small particles, travelling with high speed at grazing angles and are created during initial stage of the impact. However, they remain very near the surface and do not contribute in overall escape of the particles from Moon. Similarly, the spalls produced during the impact travel vertically upward and fall down on the surface, subsequently. The material ejected from regolith contains few, if any, spalls and mainly consists of dust grains of the regolith itself (Gault et al., 1963). In our study of the escape of material from Moon, we consider only 7

Journal Pre-proof a portion of the ejected material, which is in the cone shaped zone and we call it as the ejecta. We neglect the jets and spalls in the computation, because they do not contribute in the escape from Moon due to their very small vertical velocity. The impact theory for Moon is established to understand characteristics of the ejecta coming out from the lunar surface due to micrometeorite impacts. We have used the impact theory to compute the parameters of ejecta like total ejected mass, ejecta size, ejecta velocity and number of ejected particles.

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Incoming Micrometeorites (Mass and Velocity)

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Ejecta (Escape for  > 2.39 km/s)

Jet

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Spall

Jet

Central Pit

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Fine Grain Regolith

Lunar Surface

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Figure 3: Micrometeorite impact process on lunar surface

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Gade and Miller (2015) have reported estimation of the ejecta based on input details of the incoming particles. According to their report, the target material is finely commuted in fine solid fragments and the fragments are ejected in a thin debris cone, after the crater formation due to an impact. It is expected that physical state of the ejected material is solid and the ejection angle is around 60°-80° from the lunar surface for a normal impact, while the ejection velocity is from a few m/s to a few km/s given as inversely proportional to the fragment size (Gade and Miller, 2015). For the ejecta created due to an impact on lunar surface, the total ejected mass is given as (Gault et al., 1963) 𝑚𝑒𝑗 = 𝛽 × 𝐶 × 0.5 𝑚𝑖 𝜈𝑖 2 𝑐𝑜𝑠 2 𝜃𝑖 = 𝛾 𝑚𝑖

and

𝑚𝑒𝑗 = 𝛽 × 𝐶 × 0.5 𝑚𝑖 𝜈𝑖 2 𝑐𝑜𝑠 2 60 = 𝛾 𝑚𝑖

for 𝜃𝑖 ≤ 60° for 𝜃 > 60° (3)

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Journal Pre-proof where 𝜃𝑖 = incidence angle with respect to normal of lunar surface, 𝑚𝑒𝑗 = total ejected mass, 𝜈𝑖 = incoming micrometeorite velocity, 𝑚𝑖 = incoming micrometeorite mass, 𝛽 = 1 for 𝑑𝑝 < 1 µm and 0.4 for 𝑑𝑝 > 100 µm (with linear interpolation in between) as given by (Rival and Mandeville, 1999), 𝑑𝑝 = diameter of incoming projectile, 𝐶 = 1.5  10-5 for quartz bearing soil surface and 𝐶 = 4.9  10-5 for ice bearing surface. The value of 𝐶 for water ice mixed with the regolith is between the two values given above and it is useful for the permanently shadowed region of Moon (described in later part

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of this article). Further, the maximum size of ejecta released during hypervelocity impact is given by Rival and Mandeville (1999) as 𝛿𝑚𝑎𝑥 , the mass of largest fragment as 𝑚𝑏 and the

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minimum size of fragment released is 𝛿𝑚𝑖𝑛 = 0.1 µm. In addition, the minimum velocity of ejecta is given as 𝜈𝑚𝑖𝑛 = 10 m/s, while the maximum velocity of ejecta is 𝜈𝑚𝑎𝑥 = 𝜈𝑖 for

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normal incidence (Rival and Mandeville, 1999) on lunar surface. The density of loose

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regolith target material on the lunar surface is 𝜌𝑡 = 1.5 g cm-3 as reported by Heiken et al. () as

𝑣𝑖 −10 ×𝛿𝑚𝑎𝑥 𝛿𝑚𝑖𝑛 𝛿𝑚𝑎𝑥 −𝛿𝑚𝑖𝑛

𝛿

+

10𝛿𝑚𝑎𝑥 −𝑣𝑖 𝛿𝑚𝑖𝑛 𝛿𝑚𝑎𝑥 −𝛿𝑚𝑖𝑛

(4)

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𝑣𝑒𝑗 (𝛿) =

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(1991). Rival and Mandeville (1999) have provided the velocity of ejecta as a function of size

We have divided the minimum (𝛿𝑚𝑖𝑛 ) and maximum (𝛿𝑚𝑎𝑥 ) ejecta size (or diameter) into a number of equal size bins and taken the average value in each bin as a representative size within that bin. The details of the selection of number of bins are given below. Further, Juhasz et al., (1993) have reported the number of ejected particles in the size range (𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ) due to an impact of a single particle on the lunar surface as 3 𝛾 (4−𝑞) 𝑄 ) 𝑡 (𝑞−1)

𝑁𝑒𝑗 (𝑚) = (4 𝜋 𝜌

with 𝑄 =

𝑞−1 3

(𝑎𝑚𝑖𝑛1−𝑞 − 𝑎𝑚𝑎𝑥 1−𝑞 ) 𝑚𝑄

(5)

and 𝛾 is given in Equation (3). Juhasz et al. (1993) have reported 𝛾 = 1700

based on the observations of meteoroids, which have average impact velocity of 15 km/s. In our work, we use Equation (3) for finding the values of 𝛾 by varying the velocity as well as the incidence angle of incoming projectile. 9

Journal Pre-proof The parameter 𝑞 is measured from impact experiments and it ranges from 2 .75 to 3.5 (Asada, 1985; Grun et al., 1980). In the present work, the initial results are based on a value of 𝑞 as 2.75, however, we have compared the effect of change in the parameter 𝑞 towards the end. Based on a given value of 𝑞, one needs to consider some number of bins in the analysis. For the selected number of bins, it is essential to have the summation of ejected mass multiplied by ejected particles (∑ 𝑚𝑒𝑗 𝑛𝑒𝑗 ) for all the selected mass bins to be less than or equal to the total ejected mass (𝑀𝑒𝑗 ) in the cone. We found the minimum number of bins to be 150 for 𝑞 = 2.75 and it is 1400 for 𝑞 = 3.5 to satisfy this condition. We have therefore, used the minimum number of bins as 1500 in all the analysis. The difference between the

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quantity ∑ 𝑚𝑒𝑗 𝑛𝑒𝑗 and the quantity 𝑀𝑒𝑗 could be due to the consideration of discrete nature

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4. Estimation of Particle Escape from Moon

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in the ejected mass (taken as bins).

We consider ejected particles in the range from 10-21 to 10-1 kg and obtain the number

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of ejected particles due to a single impact event on the lunar surface using Equation (5).

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Initially, we have considered the normal incidence of incoming projectile and later on, we have also checked the effect of incidence angle mentioned in the later part. The results are

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depicted in Figure 4, where upper plot is for the number of ejected particles, middle plot is for the mass of ejected particles and lower plot is for the velocity of ejected particles, against the incoming projectile mass. It can be seen from Figure 4 that the points are clustered and

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each point corresponds to the selected bin in the total fifteen hundred bins. In the lower plot of Figure 4, a horizontal (red) line is drawn at the escape velocity of Moon (i.e., 2.39 km/s) and it serves as a threshold for deciding the escape from Moon. Those particles whose vertical component of velocity is above the threshold, will escape from Moon.

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Figure 4: Logarithmic plot of number (red, circle) of ejected particles, mass (blue, asterisk) of ejected particles and velocity (green, cross) of ejected particles versus mass of the normally incident projectile. Each value of

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incoming particle gives fifteen hundred points, taken as number of bins and the points are clustered due to

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nearby values. The threshold in velocity is also shown as straight (red) line in the lower plot.

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The ejected particles may spread in various directions, after the impact, in the horizontal as well as vertical planes. For a normal incidence, the distribution in azimuth is

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uniform, while for other incident angles, the distribution in azimuth could be non-uniform. However, for the escape study in the present work, the horizontal plane distribution is not of

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significance and therefore, we have omitted it. For the zenith distribution of ejected particles,

ℎ(𝜃𝑒𝑗 ) = where 𝜎 =

𝜃𝑚𝑎𝑥 =

5 6

𝜋 60

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the Gaussian function is given by (Gade and Miller, 2015, Rival and Mandeville, 1999)

1

𝜎 √2

exp (− 𝜋

(𝜃𝑒𝑗 −𝜃𝑚𝑎𝑥 ) 2 𝜎2

2

)

for 0 ≤ 𝜃 ≤ 2 𝜋

(6)

and

𝜃𝑖 +

𝜋 6

for 𝜃𝑖 ≤ 60° while 𝜃𝑚𝑎𝑥 =

4𝜋 9

for 𝜃𝑖 > 60° (Gade and Miller, 2015)

Considering the zenith variation given in Equation (6), we obtained the results as shown in Figure 5, as the number of ejected particles versus mass of incoming projectile and zenith angle of the ejected particles. These results are important to study the escape of particles. This is because the particles travelling faster but very close to the lunar surface will land back on the surface and will have no chance of escaping the lunar gravity. One can observe from Figure 5 that the zenith angular distribution is similar to Gaussian, peaking at around 30 o

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in a particular zenith direction from Figure 5 due to an impact on the lunar surface.

Figure 5: Logarithm of number of ejected particles versus zenith angle of ejected particles and logarithm of

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incoming projectile mass.

From the results in Figure 4, we have computed the total ejected mass as well as total

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escaping mass for single impact (after considering the threshold of vertical component of velocity) as a function of incoming projectile mass. The results are depicted in upper and

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middle plots of Figure 6. Now, the incoming flux at Moon is given in Figure 1 and adding its contribution, we can obtain the effective escaping mass flux from Moon as shown in lower

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plot of Figure 6. One can observe from Figure 6 that total escaping mass (for single impact) and escaping mass flux from Moon are only present for few values of the incoming projectile mass. For higher values of the incoming projectile mass (mi > 10-12 kg), there is no escape due to larger fragment sizes and lower velocities of the ejected particles.

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Figure 6: Logarithmic plot of (a) total ejected mass due to single event, (b) total escaping mass due to single

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event and (c) escaping mass flux for Moon versus logarithm of incoming projectile mass.

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In order to understand practical scenario on Moon, we need to consider expected distribution of incoming dust particles with respect to the incident angle. Hughes (1993) has

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reported probable angle distribution for incoming projectiles. We have computed the escaping mass rate for Moon using 𝑞 = 2.75 as well as 𝑞 = 3.5 for various incident angles,

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whose results are depicted in Figure 7 and Figure 8, respectively. One can observe from Figure 7 and Figure 8 that for each incident angle, the corresponding plot is slightly different.

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Furthermore, to understand the practical case on Moon, we have taken the weighted average

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of all seven plots in Figure 7 (and Figure 8) using

𝑔(𝑚𝑖 ) =

𝑤1 𝑓1 (𝑚𝑖 )+𝑤2 𝑓2 (𝑚𝑖 )+⋯+𝑤7 𝑓7 (𝑚𝑖 ) 𝑤1 +𝑤2 +⋯+𝑤7

(7)

where the weights 𝑤1 , 𝑤2 , … , 𝑤7 are taken according to the angle distribution reported by Hughes (1993). The weighted average in Equation (7) has been taken from absolute values (i.e., before taking logarithm) of parameters in Figure 7 (and Figure 8). The function 𝑔(𝑚𝑖 ) in Equation (7) is called mass rate (MR), which is used in our computations and plots. Table 1 gives the incoming projectile angle and its arrival probability (Hughes, 1993) or weight (𝑤𝑖 ) used in Equation (7). Table 1: Incoming projectile angle at which the projectile strikes the lunar surface with respect to surface normal and its arrival probability or weights (𝑤𝑖 ) used in Equation (7) (Hughes, 1993)

Incoming Projectile Angle

Projectile Arrival Probability 13

Journal Pre-proof (Deg) 0 15 30 45 60 75 90

(%) 0.154 4.385 7.541 8.668 7.541 4.385 0.154

After taking the weights (𝑤𝑖 ) from Table 1, we obtained the resultant weighted average, escaping mass rates of Moon for 𝑞 = 2.75 as well as 𝑞 = 3.5 as shown in Figure 9.

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It should be noted that there will be some error due to the weighted averaging procedure. In

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addition, there can be error in the measurements of incoming particle velocity and mass by the Galileo dust detector. The error factors are ten and two for the measured mass and

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measured velocity respectively, as reported by Grun et al. (PSS, 1995). We have incorporated the measurement errors in all the computations. Further, the errors due to measurement as

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well as the weighted sum are taken together in the escape rate as shown by error bars in

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Figure 9. The error bars in Figure 9 correspond to the maximum error for a given incoming projectile mass. It should be noted that this escaping mass rate is independent of incoming

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projectile angles, as we have generalised it over the same.

Figure 7: Logarithmic plot of escaping mass rate for Moon versus logarithm of incoming projectile mass for 𝑞 = 2.75 and various incident angles 𝜃𝑖 .

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Figure 8: Logarithmic plot of escaping mass rate for Moon versus logarithm of incoming projectile mass for

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𝑞 = 3.5 and various incident angles 𝜃𝑖 .

Figure 9: Logarithmic plot of weighted average of escaping mass rate for Moon versus logarithm of incoming projectile mass for 𝑞 = 2.75 and 𝑞 = 3.5, normalized for various incident angles 𝜃𝑖 with error bar.

From the results in Figure 9, we get the equations of weighted average escaping mass rates for regolith. We have obtained similar equations for water ice as well. All the derived equations of mass escape rates are as follows. 𝑔(𝑚𝑖 ) = −(3.997 × 10−4 ) ∙ (log 𝑚𝑖 )4 − 0.028 (log 𝑚𝑖 )3 − 0.727 (log 𝑚𝑖 )2 − 8.487 log 𝑚𝑖 − 42.896

(for regolith target with 𝑞 = 2.75)

15

(8)

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𝑔(𝑚𝑖 ) = −(6.696 × 10−4 ) ∙ (log 𝑚𝑖 )4 − 0.046 (log 𝑚𝑖 )3 − 1.148 (log 𝑚𝑖 )2 − 12.512 log 𝑚𝑖 − 54.580

(for regolith target with 𝑞 = 3.5)

(9)

𝑔(𝑚𝑖 ) = −(4.852 × 10−4 ) ∙ (log 𝑚𝑖 )4 − 0.033 (log 𝑚𝑖 )3 − 0.807 (log 𝑚𝑖 )2 − 8.983 log 𝑚𝑖 − 43.161

(for ice target with 𝑞 = 2.75)

(10)

and

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𝑔(𝑚𝑖 ) = −(6.466 × 10−4 ) ∙ (log 𝑚𝑖 )4 − 0.044 (log 𝑚𝑖 )3 − 1.106 (log 𝑚𝑖 )2 − (for ice target with 𝑞 = 3.5)

(11)

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11.994 log 𝑚𝑖 − 51.713

water ice present on the lunar surface.

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Using Equations (8) to (11), one can obtain the escape rates for any proportion of regolith and

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5. Water Ice Bearing Permanently Shadowed Region

It is expected that water ice is trapped in PSR regions near the poles of Moon. It

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occupies 12866 km2 and 16055 km2 in the North and South polar regions, respectively, as reported by Mazarico et al. (2011). When compared to total surface area of Moon this

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amounts to 0.0762 %. From the total PSR, ~3.5 % of cold traps exhibits ice exposure (Li et al., 2018). For this (3.5 %) fraction of the PSR, we have considered pure ice as the target

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material. Further, the fraction of water ice mixed with the lunar regolith is ~30 % as suggested by Thomson et al. (2012) in the Shackleton crater, which was revealed by LRO Mini-RF orbital radar. For the remaining 96.5 % of the PSR we assume 30 % water ice mixed with the regolith, and such target is described by two component mixing model for the constant 𝐶 in Equation (3) 𝐶𝑃𝑆𝑅 = 𝑓𝑟𝑒𝑔 𝐶𝑟𝑒𝑔 + 𝑓𝑖𝑐𝑒 𝐶𝑖𝑐𝑒

(12)

where 𝐶𝑃𝑆𝑅 is the constant for PSR, 𝑓𝑟𝑒𝑔 and 𝑓𝑖𝑐𝑒 are fractions of regolith and water ice in the mixture respectively, while 𝐶𝑟𝑒𝑔 and 𝐶𝑖𝑐𝑒 are constants for regolith and water ice target respectively. This provides the value of 𝐶𝑃𝑆𝑅 as 2.52  10-5 for 30 % water ice, mixed with the lunar regolith. Thus, for the estimation of water ice escape, we have used 3.5 % of the

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Journal Pre-proof PSR as a target with ice and 96.5 % of the PSR as a target with mixture of water ice and regolith. To obtain the overall mass balance on Moon, we have compared the incoming mass rate and escaping mass rate for 𝑞 = 2.75 as well as 𝑞 = 3.5 and the results are shown in Figure 10. One can observe from Figure 10 that there is slight change in the plots for the two cases (i.e., 𝑞 = 2.75 and 𝑞 = 3.5) and practically, the escape rates will fall between the two plots. Along with the regolith escape rate, we have also derived the mass escape rates for the mixture (of regolith and water ice) and for the water ice in the PSR, as per the proportion given above. These results are shown in Figure 10. It should be noted that the mass escape

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rates in Figure 10 are for the case when whole surface or target is made up of either only

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regolith or mixture or only ice. Further, using the results in Figure 10, we have computed the actual escape rates for the case of Moon, after considering the distribution of target materials

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like regolith, water ice or their mixture. It should also be noted that the effective or total escape for the regolith is due to the contribution by regolith target as well as the mixed target.

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Similarly, the effective or total escape for the water ice is due to the contribution from water

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ice surface as well as the mixture. This is described as

(13)

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𝐸𝑅 = 𝑝𝑚𝑖𝑥 𝐸𝑅𝑚𝑖𝑥 + 𝑝𝑝𝑢𝑟𝑒 𝐸𝑅𝑝𝑢𝑟𝑒

where 𝐸𝑅 is the effective or total escape rate (for regolith or water ice), 𝑝𝑚𝑖𝑥 is the proportion

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of water ice mixed with regolith (i.e., 30 % water ice and 70 % regolith in the 96.5 % of PSR), 𝐸𝑅𝑚𝑖𝑥 is the escape rate of the material (i.e., regolith or water ice) from the mixed target (which is 96.5 % of PSR), 𝑝𝑝𝑢𝑟𝑒 is the proportion of the material (i.e., regolith or water ice) alone (i.e., 3.5 % of PSR for the water ice or non-PSR for the regolith), while 𝐸𝑅𝑝𝑢𝑟𝑒 is the escape rate from this pure (either regolith or water ice) target. Using Equations (8) to (11), we have found effective or total mass escape rate of regolith as well as that of water ice for the Moon and the results are depicted in Figure 11. One can observe from Figure 11 that the escape rates (of regolith or water ice) will remain between the two curves given for 𝑞 equal to 2.75 and 3.5, which represents the possible diversity in the target. Further, the mass escape rate is dominated by the lower incoming projectile mass and it is ~3.977 × 10-5 [2.410 × 10-5, 40.580 × 10-5 ] kg/s for regolith and it is ~6.087 × 10-8 [2.295 × 10-8, 33.121 × 10-8] kg/s for water ice. The quantities in the square

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Journal Pre-proof brackets are upper and lower limits of the given parameter, which have been arrived at, after

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considering all the errors mentioned earlier.

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Figure 10: Incoming and escaping mass rates of Moon for two cases (i.e., 𝑞 = 2.75 as well as 𝑞 = 3.5).

Figure 11: Total regolith and water ice (volatile) escape rates for Moon for two cases (i.e., 𝑞 = 2.75 as well as 𝑞 = 3.5).

Finally, we have found the individual mass escape rates for each incoming particle mass, the average mass escape rates and the total mass escape rates for Moon and given in Table 2. We found that on an average, the regolith escapes at a rate of ~1.848 × 10-5 [1.386 × 10-5, 8.527 × 10-5] kg/s and its total escape from Moon is ~2.218 × 10-4 [1.662 × 10-4, 10.232 × 10-4 ] kg/s due to micrometeorite bombardment on the lunar surface. Similarly, on an average,

the water ice escapes at a rate of ~1.657 × 10-8 [1.302 × 10-8, 6.306 × 10-8] kg/s and its total 18

Journal Pre-proof escape from Moon is ~1.988 × 10-7 [1.562 × 10-7, 7.567 × 10-7 ] kg/s due to the bombardment. One can see from Table 2 that Moon receives the dust particles, which is at least an order higher than those escaping and going into the outer space. It is interesting to note that water ice (volatile) escape due to micrometeorite bombardment on the lunar surface is ~ four orders less than the incoming dust particles. The water ice on Moon is an important source of energy for future missions. Though, the amount of water ice on Moon is less, its small escape could become significant over a period of time, and cause the depletion.

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Table 2: Summary of incoming projectile mass (first column), incoming mass rate on Moon (second column), escaping mass rate for 𝑞 = 2.75 (third column) and escaping mass rate for 𝑞 = 3.5 (fourth column). The escaping

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mass rate for regolith and ice are the effective or total values, after taking contributions from only regolith (excluding the PSR), the mixture of ice and regolith in 30:70 proportion (part of the PSR) as well as only ice

-20

-7.837

-19

-7.648

-4.565

-7.620

-4.668

-7.626

-18

-7.394

-4.711

-7.863

-4.727

-7.774

-17

-7.019

-4.849

-8.086

-4.745

-7.865

-6.501

-4.979

-8.282

-4.716

-7.892

-5.884

-5.110

-8.458

-4.649

-7.864

-5.247

-5.261

-8.634

-4.568

-7.811

-13

-4.616

-5.460

-8.847

-4.517

-7.778

-12

-3.942

-5.747

-9.145

-4.551

-7.835

-11

-3.348

-6.167

-9.587

-4.746

-8.066

-10

-2.941

-6.780

-10.246

-5.190

-8.572

-09

-2.748

-

-

-

-

-08

-2.755

-

-

-

-

-07

-2.906

-

-

-

-

-06

-3.145

-

-

-

-

-05

-3.433

-

-

-

-

-04

-3.747

-

-

-

-

-03

-4.074

-

-

-

-

-02

-4.408

-

-

-

-

-01

-4.745

-

-

-

-

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Logarithm of Escaping Mass Rate for q = 2.75 (kg/s)

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Logarithm of Incoming Mass Rate (kg/s) -8.001

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Logarithm of Escaping Mass Rate for q = 3.5 (kg/s)

Regolith

Ice

Regolith

Ice

-4.299

-7.161

-4.533

-7.278

-4.421

-7.375

-4.590

-7.447

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Logarithm of Incoming Projectile Mass (kg)

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(part of the PSR). All values are given in logarithmic scale.

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-3.423

-4.827

-7.847

-4.656

-7.723

Total

-2.100

-3.748

-6.768

-3.577

-6.644

Conclusion and Implications A continuous supply of micrometeorites on Moon causes ejecta to come out and leads some of the particles to escape the planetary body. We have used existing model of micrometeorite flux on Moon. To consider a practical scenario, Galileo observations of the IDPs near Moon are used to obtain the variation in projectile mass and velocity, along with their errors. The impact theory is applied to estimate total ejected mass and total escaping

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mass of the regolith, for a wide range of incoming micrometeorite particles. Further, escaping mass flux is derived based on a threshold in the velocity of ejecta particles. The results are

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compared for various angles of incidence of the incoming projectiles and a weighted average of the escaping mass rate is suggested. We found upper limit of regolith escape rate as ~2.218

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× 10-4 [1.662 × 10-4, 10.232 × 10-4 ] kg/s and that of water ice escape rate as ~1.988 × 10-7 [1.562 ×

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10-7, 7.567 × 10-7 ] kg/s. Further, Moon receives the dust particles, which is one order higher

than that lost due to the escape and it indicates that Moon is gradually becoming heavier.

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From our findings, we get the water ice escape rate to be ~6.271 [4.926, 23.863] kg/year. Though slowly, Moon could be depleted of water ice resource over a period of time due to

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micrometeorite impact on the surface. The water ice is a useful source for future manned mission, habitation and as a possible source of hydrogen (as a fuel) for the rockets. The

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results presented in this paper are useful to understand dust and volatile escape from Moon. In addition, the approach in this paper can also be applied, on a similar line, to other planetary bodies in the solar system, for understanding the escape process.

References Asada, N., “Fine fragments in high-velocity impact experiments”, J. Geophys. Res., 90, 12,445-12,453, 1985. Banaszkiewicz, M., and Ip, W. H., A statistical study of impact ejecta distribution around Phobos and Deimos, Icarus, 90, 237-253, 1991. Berg, O. E. and Grun, E., “Evidence of hyperbolic cosmic dust particles”, Space Research 13, 1047-1055, 1973. Canup, R. M., Visscher, C., Salmon, J. and Fegley Jr, B., “Lunar volatile depletion due to incomplete accretion within an impact-generated disk”, Nature Geoscience, Vol., 8, pp. 918-921, 2015. Davis, D. R., Housen, K. R. and Greenberg, R., “The unusual dynamical environment of Phobos and Deimos”, Icarus, Vol. 47, Issue 2, pp. 220-233, 1981. Desch, S. J. and Taylor, G. J., “A Model of the Moon's Volatile Depletion”, 42 nd LPSC, The Woodlands, Texas, #2005, 2011.

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Journal Pre-proof Farrell, W. M., Hurley, D. M. and Zimmerman, M. I., “Spillage of lunar polar crater volatiles onto adjacent terrains: The case for dynamic processes”, Geophys. Res. Lett., vol. 42, pp. 3160-3165, 2015. doi: 10.1002/2015GL063200. Francesconi, A. et al., “Characterization of ejecta from HVI on spacecraft outer surfaces”, IADC-11-05, 2013. Online

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05,%20Characterization%20of%20Ejecta%20from%20HVI.pdf Juhasz, A., Tatrallyay M., Gevai, G. and Horanyi, M., “On the density of the dust halo around Mars”, Journal of Geophysical Research, Vol. 98, No. E1, pp. 1205-1211, 1993. Gade, A. and Miller, A., “PC Version of DEBRIS Impact Analysis Tool”, ESABASE2/Debris Release 7.0 Technical

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Grun, E. and Zook, H. A., “Dynamics of micrometeoroids in the inner Solar System” In Solid Particles in the

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Grun, E., Zook, H. A., Fechtig, H. and Giese, R. H., “Collisional balance of the meteoritic complex”, Icarus, 62,

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1995. https://doi.org/10.1016/0032-0633(94)00234-I Halekas, J. S., Bale, S. D., Mitchell, D. L. and Lin, R. P., Electrons and magnetic fields in the lunar plasma wake. J. Geophys. Res., 2005, 110, A07222.

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Halekas, J. S. et al., Extreme lunar surface charging during space weather events. Geophys. Res. Lett., 34, L02111, 2007.

Heiken, G. H., Vaniman, D. T. and French, B. M., Lunar Source Book, Cambridge University Press, 1991. Hoffmann, H. J., Fechtig, H., Grun, E., and Kissel, J., “First results of the micrometeoroid experiment S215 on HEOS 2 Satellile”, Planegtary and Space Science, 23, 215-224, 1975a. Hoffmann, H. J., Fechtig, H., Grun, E., and Kissel, J., “Temporal fluctuations and anisotropy of the micrometeoroid flux in the Earth-Moon system”, Planegtary and Space Science, 23, 985-991, 1975b. Horanyi, M., Charged dust dynamics in thesolar system. Annu. Rev. Astron. Astrophys., 1996, 34, 383–418. Horanyi, M., Szalay, J. R., Kempf, S., Schmidt, J., Grun, E., Srama, R. and Sternovsky, Z., “A permanent, asymmetric dust cloud around the Moon”, Nature, Vol. 522, pp. 324-326, 2015. Hughes, D. W., “Meteorite incidence angles”, Journal of British Astronomical Association, Vol. 103, No. 3, pp. 123-126, 1993. Ishimoto, H., “Formation of Phobos/Deimos Dust Rings”, Icarus, 122, 153-165, 1996. Krivov, A. V. and D. P. Hamilton, “Martian dust belts: waiting for discovery”, Icarus, Vol. 128, pp. 335–353, 1997.

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Space (ed. Grard, R. J. L.), D. Reidel Publishing Co, Dordrecht, The Netherlands, 1973, pp. 347–361. Mazarico, E., Neumann, G. A., Smith, D. E., Zuber, M. T. and Torrence, M. H., “Illumination conditions of the

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Naumann, R. J., “The near Earth Meteoroid Environment”, NASA TND 3717, 1966. Pabari, J. P. and Banerjee, D., “Levitation of charged dust grains and its implications in lunar environment”,

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Current Science, Vol. 110, No. 10, 25, pp. 1984-1989, 2016. Pabari, J. P. and Bhalodi, P. J., “Estimation of micrometeorites and satellite dust flux surrounding Mars in the light of MAVEN results”, Icarus, Elsevier, Vol. 288, May 2017, pp. 1-9. Pabari, J. P., Bhalodi, P. J. and Patel, D. K., “Mars Orbit Dust Experiment (MODEX) for future Mars orbiter”, 47th Lunar and Planetary Science Conference, The Woodlands, Texas, #1419, 2016. Rival, M. and Mandeville, J. C., “Modeling of Ejecta Produced upon Hypervelocity Impacts”, Space Debris, Vol. 1, Issue 1, pp.45-57, 1999. Schneider, E., “Impact ejecta exceeding lunar escape velocity”, The Moon, Vol. 13, pp. 173-184, 1975. Stubbs, T. J., Vondrak, R. R. and Farrell, W. M., “A dynamic fountain model for lunar dust”, Advances in Space Reseasrch, 37, 59-66, 2006. Thomson, B. J., Bussey, D. B. J., Neish, C. D., Cahill, J. T. S., Heggy, E., Kirk, R. L. et al., “An upper limit for ice in Shackleton crater as revealed by LRO Mini‐RF orbital radar”, Geophys. Res. Lett., 39, L14201, 2012.

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Journal Pre-proof Highlights  Existing theory is applied to the lunar surface for understanding regolith and water ice (volatile) escape due to impact of hypervelocity micrometeorites. For incoming micrometeorites in the mass range from 10-21 to 10-1 kg, equations of mass escape rates are suggested for regolith as well as water ice (volatile).



The upper limit of escape rates are found to be ~2.218 × 10-4 kg/s and ~1.988 × 10-7 kg/s for regolith and water ice (volatile), respectively.



Moon loses its water ice (volatile) at a rate of ~6.271 kg/year.

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

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