Effect of the Earth’s compression on the physical libration of the Moon

The effect of the Earth??s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 10⁶ than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon??s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon??s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth??s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth??s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L _? = 90° or L _? = 270°. The famous Cassini??s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth??s compression on the Moon??s libration in longitude, a harmonic with an amplitude of 0.03?? and period of T ₈ = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth??s compression on the variation in spin angular velocity of the Moon proves to be negligible.     

Effects of a physical librations of the moon caused by a liquid core,and determination of the fourth mode of a free libration

An analytical theory of lunar physical librations based on its two-layer model consisting of a non-spherical solid mantle and ellipsoidal liquid core is developed. The Moon moves on a high-precision orbit in the gravitational field of the Earth and other celestial bodies. The defined fourth mode of a free libration is caused by the influence of the liquid core, with a long period of 205.7 yr, with amplitude S = 0″0395 and with an initial phase Π₀ = ?134° (for the initial epoch 2000.0). Estimates of dynamic (meridional) oblatenesses of a liquid core of the Moon have been estimated: ?_D = 4.42 × 10^?4, μ_D = 2.83 × 10^?4 (?_D + μ_D = 7.24 × 10^?4). These results have been obtained as a result of comparison of the developed analytical theory of physical librations of the Moon with the empirical theory of librations of the Moon constructed on the basis of laser observations.     

On the thermal history of a moon of fission origin

Model calculations show that the thermal history of a Moon which originated by fission from the proto-Earth is the same as that for the Moon as it is currently understood. In particular, a fissioned Moon currently has a small percent of partial melt or at least near solidus temperatures below depths of 800 km in accord with the seismic data which show that the deep interior of the Moon has a very lowQ. The models have moderate (20–50%) degrees of partial melting in the upper mantle (depths < 300 or 200 km) in the period between 3 to 4 × 10⁹ years ago and, therefore, can account for the mare filling epoch. Finally the heat flow of the models is 18 ergs cm^–2 s^–1 which is close to the average of 19 ergs cm^–2 s^–1 derived from the Apollo heat flow experiments. These findings add further support for the fission origin of the Moon.     

Recent results on the mass,gravitational field and moments of inertia of the moon

Doppler tracking data from the Lunar Orbiter series of spacecraft have been used in a more complete analysis of the spherical harmonic coefficients of the lunar gravitational field through thirteenth degree and order. The value obtained for the mass of the Moon,GM = 4902.84 km³ s^–2, is in good agreement with previous results and with results obtained by alternate procedures. Acceleration contour plots, derived from the gravitational coefficients, show correlations with surface features on the near side of the Moon, but are of questionable validity for the far side because of the lack of direct tracking data on the far side. Based on the most recent gravitational field data, the current estimate for the polar moment of inertia of the Moon isC/Ma ² = 0.4019_-0.002 ^+0.004. This value indicates that the interior of the Moon can be homogeneous, but some results presented strongly suggest that the Moon is differentiated, with an excess of mass in the direction toward the Earth.Paper presented at the NATO Advanced Study Institute on Lunar Studies, Patras, Greece, September, 1971.     

On the petrology and structure of a gravitionally differentiated moon of fission origin

In a previous paper, it was shown that the basic properties and the developmental history of a gravitationally differentiated Moon of fission origin match those known for the Moon. In the first part of this report, the models of a differentiated Moon are critically reviewed based on second order considerations of some of the chemical systems used to develope the earlier models and based on new lunar data. As a result, slightly updated models are developed and the results indicate that a Moon of fission origin has a feldspar rich crust (≈70% Or_0.8Ab_5.3An_93.9 with ≈30% pyroxene and olivine) reaching an average depth of ≈65 km. A KREEP rich layer is located at the interface of the crust and the upper mantle. The upper mantle consists of peridotite (≈80% Wo₁₀En₇₀Fs₂₀ and ≈20% Fo_75–80 with ≈3% Al₂O₃ and ≈ 2% TiO₂) and reaches a depth of 300–400 km. Below 300–400 km lies a dunite (≈Fo₉₅) lower mantle. A simple model for the distribution of K, U and Th (and by inference, KREEP) in the differentiated Moon model is developed using a distribution coefficient of 0.1 for the three elements. This coefficient is derived from published data on the distribution of U in Apollo 11 basalts. The simple model successfully accounts for the observed K, U and Th contents of the various mare basalts and upland rocks and yields a heat flow of 21 erg cm^?2s^?1 for the Moon. A model for the fine structure of the peridotite upper mantle of the model Moon is developed based on the TiO₂ and trace element variations observed in the various mare basalts. It is proposed that the upper mantle is rhythmically banded on the scale of 10's of km and that this banding leads to local variations of a factor of ±3 in the K, U and Th content, _-10 ⁺⁵ in the TiO₂ content and _-∞ ⁺² in the olivine content of the peridotite. It is also proposed that this banding leads to large scale horizontal inhomogenuities in the composition of the upper mantle. It is also shown that the formation of the primitive suite of upland rocks is easily explained by the cumulation of plagioclase, which carried varying amounts of pyroxene, olivine and melt with it, during the peritectic crystallization of the last 20% of the differentiating Moon. It is found that the 100 Mg/(Mg+Fe) ratios of the mafics and the An contents of the plagioclases of the rocks are controlled by several factors, the most important of which is the ratio of melt to crystals which together formed the various upland rocks. The inverse relationship between the An contents and the Mg contents of the upland rocks is a direct consequence of the differentiation sequence proposed. The results and models presented in this paper further support the hypothesis that the Moon formed as a result of fission from the proto-Earth.     

Volatile retention from cometary impacts on the Moon

Impacts of comets and asteroids play an important role in volatile delivery on the Moon. We use a novel method for tracking vapor masses that reach escape velocity in hydrocode simulations of cometary impacts to explore the effects of volatile retention. We model impacts on the Moon to find the mass of vapor plume gravitationally trapped on the Moon as a function of impact velocity. We apply this result to the impactor velocity distribution and find that the total impactor mass retained on the Moon is approximately 6.5% of the impactor mass flux. Making reasonable assumptions about water content of comets and the comet size-frequency distribution, we derive a water flux for the Moon. After accounting for migration and stability of water ice at the poles, we estimate a total 1.3×10⁸-4.3×10⁹ metric tons of water is delivered to the Moon and remains stable at the poles over 1 Ga. A factor of 30 uncertainty in the estimated cometary impact flux is primarily responsible for this large range of values. The calculated mass of water is sufficient to account for the neutron fluxes poleward of 75° observed by Lunar Prospector. A similar analysis for water delivery to the Moon via asteroid impacts shows that asteroids provide six times more water mass via impacts than comets.     

Exposure history of lunar meteorites Queen Alexandra Range 93069 and 94269

Abstract— Cosmic-ray produced ¹⁴C (t_1/2 = 5730 years), ³⁶Cl (3.01 × 10⁵ years), ²⁶Al (7.05 × 10⁵ years), and ¹⁰Be (1.5 × 10⁶ years) in the recently discovered lunar meteorites Queen Alexandra Range 93069 (QUE 93069) and 94269 (QUE 94269) were measured by accelerator mass spectrometry. The abundance pattern of these four cosmogenic radionuclides and of noble gases indicates QUE 93069 and QUE 94269 were a paired fall and were exposed to cosmic rays near the surface of the Moon for at least several hundred million years before ejection. After the meteorite was launched from the Moon, where it had resided at a depth of 65–80 g/cm², it experienced a short transition time, ～20–50 ka, before colliding with the Earth. The terrestrial age of the meteorite is 5–10 ka. Comparison of the cosmogenic nuclide concentrations in QUE 93069/94269 and MAC 88104/88105 clearly shows that these meteorites were not ejected by a common event from the Moon.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

Orphan radiogenic noble gases in lunar breccias: evidence for planet pollution of the Sun?

Surface-correlated noble gases in lunar soils are primarily implanted SW (solar wind) noble gases. However, they also include apparently orphan radiogenic ⁴⁰Ar, ¹²⁹Xe, and ²⁴⁴Pu-derived fission Xe in excess of plausible primordial solar origin. These orphan radiogenic components are usually assigned a lunar origin, in a scenario in which radiogenic noble gases produced in the lunar interior were degassed into the transient atmosphere and then re-implanted to the lunar surface together with SW. There are some quantitative difficulties with this scenario, however, and it requires special constraints on the degassing history of the Moon that have not emerged from more general thermal history models. We therefore urge consideration of alternative hypotheses. As a possible source for the orphan radiogenic noble gases, we have examined planetary pollution of the Sun, as suggested by studies of extrasolar planetary systems (e.g., Murray et al., 2001, Astrophys. J. 555, 801-815; Israelian et al., 2001, Nature 411, 163-166). Pollution of the Sun by 2M_⊕ (two Earth mass) planetary materials (Murray et al., 2001, Astrophys. J. 555, 801-815) is likely not significant for Ar but could be important to account for orphan Xe in the Moon.     

Origin of the late heavy bombardment

Cratering data from the Moon, Mars, and Mercury have been interpreted by some as evidence that the inner solar system underwent a period of intense bombardment that ended about 4 × 10⁹ yr ago. Planetary perturbations of small objects within the solar system seem unable to account for the effect. This paper suggests that stellar perturbations from members of the original open cluster within which the Sun formed triggered the brief flux of objects responsible for the bombardment.     

The phase dependence of brightness and color of the lunar surface: a study based on integral photometric data

A procedure of an a posteriori correction of the available data on the integral photometry of the Moon is described. This procedure reduces the regular errors of the integral phase curves caused by variations of the libration parameters; the effect due to libration can reach 4%. A method allowing the integral measurements of the Moon to be compared correctly with the photometric measurements of the lunar areas or laboratory samples imitating the lunar soil has been developed. To approximate the phase curves of integral albedo in the phase-angle range from 6° to 120°, we proposed a simple empirical formula A _eq(α) = m _l e ^?ρα + m ₂ e ^?0.7α, where α is the phase angle, ρ is the factor of effective roughness, and m ₁ + m ₂ is the surface albedo at a zero phase angle. An empirical phase dependence of the slope of the lunar spectrum in the 360–1060 nm range has been obtained. The results may be used to test various theoretical models of the light scattering by the lunar surface and to calibrate the data of ground-based and space-borne spectrophotometric observations.     

A motivating exploration on lunar craters and low-energy dynamics in the Earth–Moon system

It is known that most of the craters on the surface of the Moon were created by the collision of minor bodies of the Solar System. Main Belt Asteroids, which can approach the terrestrial planets as a consequence of different types of resonance, are actually the main responsible for this phenomenon. Our aim is to investigate the impact distributions on the lunar surface that low-energy dynamics can provide. As a first approximation, we exploit the hyberbolic invariant manifolds associated with the central invariant manifold around the equilibrium point L ₂ of the Earth–Moon system within the framework of the Circular Restricted Three-Body Problem. Taking transit trajectories at several energy levels, we look for orbits intersecting the surface of the Moon and we attempt to define a relationship between longitude and latitude of arrival and lunar craters density. Then, we add the gravitational effect of the Sun by considering the Bicircular Restricted Four-Body Problem. In the former case, as main outcome, we observe a more relevant bombardment at the apex of the lunar surface, and a percentage of impact which is almost constant and whose value depends on the assumed Earth–Moon distance d_EM. In the latter, it seems that the Earth–Moon and Earth–Moon–Sun relative distances and the initial phase of the Sun θ ₀ play a crucial role on the impact distribution. The leading side focusing becomes more and more evident as d_EM decreases and there seems to exist values of θ ₀ more favorable to produce impacts with the Moon. Moreover, the presence of the Sun makes some trajectories to collide with the Earth. The corresponding quantity floats between 1 and 5 percent. As further exploration, we assume an uniform density of impact on the lunar surface, looking for the regions in the Earth–Moon neighbourhood these colliding trajectories have to come from. It turns out that low-energy ejecta originated from high-energy impacts are also responsible of the phenomenon we are considering.     

The transition from complex crater to peak-ring basin on the Moon: New observations from the Lunar Orbiter Laser Altimeter (LOLA) instrument

Impact craters on planetary bodies transition with increasing size from simple, to complex, to peak-ring basins and finally to multi-ring basins. Important to understanding the relationship between complex craters with central peaks and multi-ring basins is the analysis of protobasins (exhibiting a rim crest and interior ring plus a central peak) and peak-ring basins (exhibiting a rim crest and an interior ring). New data have permitted improved portrayal and classification of these transitional features on the Moon. We used new 128 pixel/degree gridded topographic data from the Lunar Orbiter Laser Altimeter (LOLA) instrument onboard the Lunar Reconnaissance Orbiter, combined with image mosaics, to conduct a survey of craters >50 km in diameter on the Moon and to update the existing catalogs of lunar peak-ring basins and protobasins. Our updated catalog includes 17 peak-ring basins (rim-crest diameters range from 207 km to 582 km, geometric mean = 343 km) and 3 protobasins (137-170 km, geometric mean = 157 km). Several basins inferred to be multi-ring basins in prior studies (Apollo, Moscoviense, Grimaldi, Freundlich-Sharonov, Coulomb-Sarton, and Korolev) are now classified as peak-ring basins due to their similarities with lunar peak-ring basin morphologies and absence of definitive topographic ring structures greater than two in number. We also include in our catalog 23 craters exhibiting small ring-like clusters of peaks (50-205 km, geometric mean = 81 km); one (Humboldt) exhibits a rim-crest diameter and an interior morphology that may be uniquely transitional to the process of forming peak rings. A power-law fit to ring diameters (D_ring) and rim-crest diameters (D_r) of peak-ring basins on the Moon [D_ring = 0.14 ± 0.10(D_r)^1.21±0.13] reveals a trend that is very similar to a power-law fit to peak-ring basin diameters on Mercury [D_ring = 0.25 ± 0.14(D_rim)^1.13±0.10] [Baker, D.M.H. et al. [2011]. Planet. Space Sci., in press]. Plots of ring/rim-crest ratios versus rim-crest diameters for peak-ring basins and protobasins on the Moon also reveal a continuous, nonlinear trend that is similar to trends observed for Mercury and Venus and suggest that protobasins and peak-ring basins are parts of a continuum of basin morphologies. The surface density of peak-ring basins on the Moon (4.5 × 10⁻⁷ per km²) is a factor of two less than Mercury (9.9 × 10⁻⁷ per km²), which may be a function of their widely different mean impact velocities (19.4 km/s and 42.5 km/s, respectively) and differences in peak-ring basin onset diameters. New calculations of the onset diameter for peak-ring basins on the Moon and the terrestrial planets re-affirm previous analyses that the Moon has the largest onset diameter for peak-ring basins in the inner Solar System. Comparisons of the predictions of models for the formation of peak-ring basins with the characteristics of the new basin catalog for the Moon suggest that formation and modification of an interior melt cavity and nonlinear scaling of impact melt volume with crater diameter provide important controls on the development of peak rings. In particular, a power-law model of growth of an interior melt cavity with increasing crater diameter is consistent with power-law fits to the peak-ring basin data for the Moon and Mercury. We suggest that the relationship between the depth of melting and depth of the transient cavity offers a plausible control on the onset diameter and subsequent development of peak-ring basins and also multi-ring basins, which is consistent with both planetary gravitational acceleration and mean impact velocity being important in determining the onset of basin morphological forms on the terrestrial planets.     

Quasi-critical orbits for artificial lunar satellites

We study the problem of critical inclination orbits for artificial lunar satellites, when in the lunar potential we include, besides the Keplerian term, the J ₂ and C ₂₂ terms and lunar rotation. We show that, at the fixed points of the 1-D averaged Hamiltonian, the inclination and the argument of pericenter do not remain both constant at the same time, as is the case when only the J ₂ term is taken into account. Instead, there exist quasi-critical solutions, for which the argument of pericenter librates around a constant value. These solutions are represented by smooth curves in phase space, which determine the dependence of the quasi-critical inclination on the initial nodal phase. The amplitude of libration of both argument of pericenter and inclination would be quite large for a non-rotating Moon, but is reduced to <0°.1 for both quantities, when a uniform rotation of the Moon is taken into account. The values of J ₂, C ₂₂ and the rotation rate strongly affect the quasi-critical inclination and the libration amplitude of the argument of pericenter. Examples for other celestial bodies are given, showing the dependence of the results on J ₂, C ₂₂ and rotation rate.     

The lunar mean moment of inertia and the size of the Moon's core

In this paper the relation between the uncertainty of the Moon's mean moment of inertia (I/Ma ²) and that of the core density _c is discussed with a two-layer model of the Moon - a mantle obeying Roche's law of the density distribution and a homogeneous core (Fe-core or Fe-FeS-core). When the uncertainty of I/Ma ² is 0.0023 (that is the accuracy in present observation), a core with radius of 450 km will be appropriate to the limitation of _c about 1 g cm^–3. Considering the accuracy obtained in space explorations, and the compressibility and the quasi-homogeneity of the Moon, we suggest that the parameters C ₂₀, , , a, and GM of the Moon should define as primary constants, but C ₂₂ and C/Ma ² as derived constants. Therefore, the ratio of mass of Moon to that of Earth in the IAU (1976) system of astronomical constants will become a deducible constant.     

On the origin of the moon by rotational fission

Based on simple CIPW norms for the proposed terrestrial upper mantle material, it is shown that if the Moon fissioned from the Earth and gravitationally differentiated, it could have a 72 km thick anorthosite (An₉₇) crust, a calcium poor (3.8% by weight) pyroxenite upper mantle 100 Mg/Mg + Fe = 75 to 80) ending at a depth of 313 km and a dunite (Fo_{93_95}) lower mantle below a depth of 313 km. Refinements of these simple norm models, based on the cooling history, crystallization sequence and the variations of the 100 Mg/Mg + Fe ratio of the liquid and crystals during the crystallization sequence, indicate that the final form of such a Moon could have the following properties: (1) a primitive, cumulate anorthosite - minor troctolite crust with intrusive and extrusive feldspathic basalts and KREEP rich norites; the thickness of this crust would be 75 km; (2) a zone in the bottom of the crust and the top of the upper mantle which is rich in KREEP, the incompatible elements, silica, and possibly voltiles; this zone would be the source area for the upland feldspathic basalts, KREEP rich norites and KREEP and silica rich fluids; (3) an upper mantle between the depths of 75 km and 350 to 400 km which consists of peridotite containing 80–85% pyroxene (Wo₁₀En_{68_72}Fs_{18_22}) and 15–20% olivine (Fo_{75_80}); the Al₂O₃ content of the upper mantle is 3%; the peridotite layer would be the source area for mare basalts and; (4) a lower mantle below a depth of 350–400 km which consists of dunite (Fo_{93_97}).The cooling history of such a moon indicates that the primitive anorthosite crust would have been completely formed within 10⁸ yr after fission. The extrusion and intrusion of upland basalts and KREEP rich norites and the metamorphism of the crustal rocks via KREEP and silica rich fluids would have ended about 4 × 10⁹ yr ago when cooling well below the solidus reached a depth of 150 km. As cooling continied, the only source of magmas after 4 × 10⁹ yr ago would have been the peridotite upper mantle, i.e. the source area of the mare basalts. Extrusion of mare basalts ended when cooling below the solidus reached the top of the refractory dunite lower mantle 3-3.3 × 10⁹ yr ago.Thus, it is shown that the chemistry, primary lithology, structure and developmental history of a fissioned Moon readily match those known for the real Moon. As such, the models presented in this paper strongly support the fission origin of the Moon.Guest Scientist, supported by the Alexander von Humboldt-Stiftung.Permanent Address.     

Lunar Water: A Brief Review

One of the most exciting recent developments in the field of lunar science has been the unambiguous detection of water (either as OH or H₂O) or water ice on the Moon through instruments flown on a number of orbiting spacecraft missions. At the same time, continued laboratory-based investigations of returned lunar samples by Apollo missions using high-precision, low-detection, analytical instruments have for the first time, provided the absolute abundance of water (present mostly as structurally bound OH⁻ in mineral phases) in lunar samples. These new results suggest that the Moon is not an anhydrous body, questioning conventional wisdom, and indicating the possibility of a wet lunar interior and the presence of distinct reservoirs of water on the lunar surface. However, not all recent results point to a wet Moon and it appears that the distribution of water on the Moon may be highly heterogeneous. Additionally, a number of sources are likely to have contributed to the water inventory of the Moon ranging from primordial water to meteorite-derived water ice through to the water formed during the reaction of solar-wind hydrogen with the lunar soil. Water on the Moon has implications for future astrobiological investigations as well as for generating resources in situ during future exploration of the Moon and other airless bodies in the Solar System.     

The ‘Halo’ family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem

The Halo orbits originating in the vicinities of both,L ₁ andL ₂ grow larger, but shorter in period, as they shift towards the Moon. There is in each case a narrow band of stable orbits roughly half-way to the Moon. Nearer to the Moon, the orbits are fairly well-approximated by an almost rectilinear analysis. TheL ₂ family shrinks in size as it approaches the Moon, becoming stable again shortly before penetrating the lunar surface. TheL ₁-family becomes longer and thinner as it approaches the Moon, with a second narrow band of stable orbits with perilune, however, below the lunar surface.     

Quantitative estimation of helium-3 spatial distribution in the lunar regolith layer

³He (helium-3) in the lunar regolith implanted by the solar wind is one of the most valuable resources because of its potential as a fusion fuel. The abundance of ³He in the lunar regolith is related to solar wind flux, lunar surface maturity and TiO₂ content, etc. A model of solar wind flux, which takes account of variations due to shielding of the nearside when the Moon is in the Earth's magnetotail, is used to present a global distribution of relative solar wind flux over the lunar surface. Using Clementine UV/VIS multispectral data, the global distribution of lunar surface optical maturity (OMAT) and the TiO₂ content in the lunar regolith are calculated. Based on Apollo regolith samples, a linear relation between ³He abundance and normalized solar wind flux, optical maturity, and TiO₂ content is presented. To simulate the brightness temperature of the lunar surface, which is the mission of the Chinese Chang-E project's multichannel radiometers, a global distribution of regolith layer thickness is first empirically constructed from lunar digital elevation mapping (DEM). Then an inversion approach is presented to retrieve the global regolith layer thickness. It finally yields the total amount of ³He per unit area in the lunar regolith layer, which is related to the regolith layer thickness, solar wind flux, optical maturity and TiO₂ content, etc. The global inventory of ³He is estimated as 6.50×¹⁰⁸ kg, where 3.72×¹⁰⁸ kg is for the lunar nearside and 2.78×¹⁰⁸ kg is for the lunar farside.     

The Earth-figure perturbations in the Lunar theory

We compute the perturbations on the motion of the Moon due to the shape of the Earth. The zonal terms inJ ₂,J ₃, andJ ₄ are considered. The accuracy is estimated at 3×10^–5 and the results compared with previous theories.