Density within the moon and implications for lunar composition

Density models for the Moon, including the effects of temperature and pressure, can satisfy the mass and moment of inertia of the Moon and the presence of a low density crust indicated by the seismic refraction results only if the lunar mantle is chemically or mineralogically inhomogeneous. IfC/MR ² exceeds 0.400, the inferred density of the upper mantle must be greater than that of the lower mantle at similar conditions by at least 0.1 g cm^–3 for any of the temperature profiles proposed for the lunar interior. The average mantle density lies between 3.4 and 3.5 g cm^–3, though the density of the upper mantle may be greater. The suggested density inversion is gravitationally unstable, but the implied deviatoric stresses in the mantle need be no larger than those associated with lunar gravity anomalies. UsingC/MR ³=0.400 and the recent seismic evidence suggesting a thin, high density zone beneath the crust and a partially molten core, successful density models can be found for a range of temperature profiles. Temperature distributions as cool as several inferred from the lunar electrical conductivity profile would be excluded. The density and probable seismic velocity for the bulk of the mantle are consistent with a pyroxenite composition and a 100 MgO/(MgO+FeO) molecular ratio of less than 80.Communication presented at the Lunar Science Institute Conference on Geophysical and Geochemical Exploration of the Moon and Planets, January 10–12, 1973.     

Numerical investigation of collision orbits of lunar satellites

In the present study an investigation of the collision orbits of natural satellites of the Moon (considered to be of finite dimensions) is developed, and the tendency of natural satellites of the Moon to collide on the visible or the far side of the Moon is studied. The collision course of the satellite is studied up to its impact on the lunar surface for perturbations of its initial orbit arbitrarily induced, for example, by the explosion of a meteorite. Several initial conditions regarding the position of the satellite to collide with the Moon on its near (visible) or far (invisible) side is examined in connection to the initial conditions and the direction of the motion of the satellite. The distribution of the lunar craters-originating impact of lunar satellites or celestial bodies which followed a course around the Moon and lost their stability - is examined. First, we consider the planar motion of the natural satellite and its collision on the Moon's surface without the presence of the Earth and Sun. The initial velocities of the satellite are determined in such a way so its impact on the lunar surface takes place on the visible side of the Moon. Then, we continue imparting these velocities to the satellite, but now in the presence of the Earth and Sun; and study the forementioned impacts of the satellites but now in the Earth-Moon-Satellite system influenced also by the Sun. The initial distances of the satellite are taken as the distances which have been used to compute periodic orbits in the planar restricted three-body problem (cf. Gousidou-Koutita, 1980) and its direction takes different angles with the x-axis (Earth-Moon axis). Finally, we summarise the tendency of the satellite's impact on the visible or invisible side of the Moon.     

Lunar paleotides and the origin of the Earth-Moon system

A new method for determining the early history of the Earth-Moon system is described. Called the study of lunar paleotides, it describes a method for explaining features of the remnant lunar gravity field, and the generation of the lunar mascons. A method for the determination of Earth-Moon distances compared with the radiometric ages of the maria is developed. It is shown that the Moon underwent strong anomalous gravitational tidal forces, for a durationt<10⁶yr, prior to the formation of the mascon surfaces. As these tidal forces had not been present at the time of the formation of the Moon, this shows that the Moon could not have been formed in orbit about the Earth.There are tides in the affairs of men which, taken at the flood, lead on to fortune... William Shakespeare 1564–1616     

A lunar signature in the geomagnetic ap-index

A study has been performed in an effort to detect a lunar signature in geomagnetic data. Ap-indices available for the period 1932–1972 have been used for this purpose. The data have been divided into some 500 lunar months which were superimposed in synchronism with the phase of the Moon. At or shortly after full Moon a peak appears which exceeds the average value by about 3 standard deviations. Possible explanations for the generation of geomagnetic disturbances by the Moon are given.Paper presented at the Conference on Interactions of the Interplanetary Plasma with the Modern and Ancient Moon, sponsored by the Lunar Science Institute, Houston, Texas and held at the Lake Geneva Campus of George Williams College, Wisconsin, between September 30 and October 4, 1974.     

Gas Dissipation from the Protosatellite Disk of Jupiter

We consider the dissipation of the gaseous component from the gas–dust accretion disk of Jupiter in which the Galilean satellites were formed. The thermal dissipation of hydrogen and helium is shown to be ineffective. It could ensure the loss of gas only for a low-mass disk and only if the rarefied outer layers of the disk are heated to 10⁴ K. Such a high disk temperature is not reached through Jupiter's radiation in existing models of its formation, but it could be provided by UV radiation of the early Sun after the dissipation of the protoplanetary disk. The viscous dissipation (with a viscosity parameter 10^–3 in the -disk model) related to disk accretion onto Jupiter could disperse a low-mass disk in 10⁷ years. A magnetocentrifugal mechanism, which produced a disk wind during accretion capable of carrying away 0.1 of the accreted gas mass, was probably also involved in the dispersal of the Jovian disk. Differential dispersion, with the loss of only hydrogen and helium and the retention of water vapor and heavier gases in the disk, is possible only in a low-mass disk model. We conclude that the water contained in the Galilean satellites was brought in mainly by solid planetesimals captured into the disk during mutual inelastic collisions in Jupiter's sphere of influence.     

On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon

From the observations of the gravitational field and the figure of the Moon, it is known that its center of mass (briefly COM) does not coincide with the center of figure (COF), and the line “COF/COM” is not directed to the center of the Earth, but deviates from it to the South–East. Here we study the deviation of the lunar COM to the East from the mean direction to Earth.At first, we consider the optical libration of a satellite with synchronous rotation around the planet for an observer at a point on second (empty) orbit focus. It is found that the main axis of inertia of the satellite has asymmetric nonlinear oscillations with amplitude proportional to the square of the orbit eccentricity. Given this effect, a mechanism of tidal secular evolution of the Moon’s orbit is offered that explains up to \(20\%\) of the known displacement of the lunar COM to the East. It is concluded that from the alternative—evolution of the Moon’s orbit with a decrease or increase in eccentricity—only the scenario of evolution with a monotonous increase in orbit eccentricity agrees with the displacement of lunar COM to the East. The precise calculations available confirm that now the eccentricity of the lunar orbit is actually increasing and therefore in the past it was less than its modern value, \(e = 0.0549\).To fully explain the displacement of the Moon’s COM to the East was deduced a second mechanism, which is based on the reliable effect of tidal changes in the shape of the Moon. For this purpose the differential equation which governs the process of displacement of the Moon’s COM to the East with inevitable rounding off its form in the tidal increase process of the distance between the Earth and the Moon is derived. The second mechanism not only explains the Moon’s COM displacement to the East, but it also predicts that the elongation of the lunar figure in the early epoch was significant and could reach the value \(\varepsilon\approx0.31\). Applying the theory of tidal equilibrium figures, we can estimate how close to the Earth the Moon could have formed.     

Accretional heating as the major cause of compositional differences among meteorite parent bodies,the Moon,and Earth

Accretional energy can be retained with sufficient efficiency in the outer layers of the Moon due to the considerable amount of debris falling back into large craters.Heating of meteorite parent bodies occurs mainly after their accretion, by destructive collisions. The heating was generally not sufficient to differentiate the parent bodies completely so that iron meteorites would originate from the mantle, rather than from the core of a meteorite parent body. Assuming that the Earth and Moon accreted from material of similar chemical composition, we suggest that only from the outer lunar shell is there a loss of gases and volatiles due to accretional melting. The Earth melted completely and degassing was efficient for the whole mass of the Earth leading to its ≈20% higher uncompressed mean density in comparison to the Moon. Because of its lower gravitational field, gases and volatiles escaped much more easily from the lunar atmosphere than from the terrestrial one, leading to the observed depletion in volatiles of the outer parts of the Moon.     

Distribution of lunar impacts by objects in parabolic low inclination orbits

This paper presents a computer investigation extending to the case of parabolic orbits, an earlier investigation conducted by Barricelli and Metcalfe (1969) on lunar impacts by external low eccentricity satellites as a means to interpret the asymmetric distribution of lunar maria. Parabolic orbits can be approximated by two kinds of objects:

1. High eccentricity external satellites may, near periapsis, approach the Moon with orbital velocity and other characteristics closely resembling those of a parabolic orbit.
2. Asteroids and meteoroids approaching the Earth-Moon system with a low velocity may have moved in a nearly parabolic orbit when they reached the lunar distance from the Earth at the time when the impacts which carved the lunar maria took place.

The investigation gives, therefore, not only additional information relevant to the interpretation of the distribution of lunar maria by the satellite impacts hypothesis (in this case high eccentricity ones), but also information about the alternative hypothesis (Wood, 1973) that asteroid impacts rather than satellite impacts were involved.     

Topology of induced lunar magnetic fields

Using the asymmetric theory of lunar induction derived by Schubertet al. (1973a), we have obtained the total and induced magnetic field line structure within the Moon and the diamagnetic cavity. Total field distributions are shown for orientations of the oscillating interplanetary field parallel, perpendicular and at 45° to the cavity axis. Induced field lines are shown only for the orientations of the interplanetary field parallel and orthogonal to the cavity axis. When compared with the field lines derived using the long wavelength limit of spherically symmetric vacuum induction theory, the configurations obtained using the asymmetric theory exhibit significant distortion. For all orientations of the interplanetary field, the field lines are strongly compressed on the sunlit hemisphere because of the confining solar wind pressure at the lunar surface and the exclusion of the field by the lunar core. Field line compression is also observed in the antisolar region in agreement with the experimental observations of Schubertet al. (1973b). and Smithet al. (1973). For the parallel orientation of the interplanetary field, antisolar compression is caused by cavity confinement of the induced field. For the interplanetary field perpendicular to the cavity axis there is, in addition to compression by the cavity boundary, redistribution of field lines from the sunlit to the night side. In this case field lines entering the Moon just forward of the limb pass through the lunar crust on the night side and then exit forward of the limb. This phenomenon manifests itself as a displacement of the null in the induced magnetic field at the surface sunward of the limb, in striking similarity to the magnetospheric field lines of the Earth.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973.     

Tectonics of the Venus and the early Precambrian

Analyzing the tectonics of planets and their satellites we use all the information available from the studies of the Earth and other celestial bodies such as the Moon, Mars and Mercury. An important condition in such analysis is naturally the scale of the phenomena compared. Most surface structures of Venus are known to have no direct analogues on the surface of the present Earth, with its global systems of mid-oceanic ridges, deep trenches and vast lithospheric plates. This might be due to the sharp differences in the present thermal regimes of the Earth and Venus. It has already been suggested in numerous papers that the key to the genesis of the Cytherean surficial structures must be looked for in the geodynamics of the Early Precambrian Earth.Such an approach appears very logical indeed since the rheology of the present Cytherean crust must be closer to that of the Precambrian rigid lithosphere of the Earth which is as if floating in the low-viscous asthenosphere. An attempt has therefore been made to evaluate certain elements in the tectonics of Venus through the theological properties of its crust comparing structural formation in the low-viscous layers of the Earth crust in the Early Precambrian with data on the morphology of structures on the surface of Venus.     

A co-accretional model of satellite formation

Numerical calculation of a simple accretion model including the effects of tidal friction indicate that coformation is tenable only if the planet's Q is less than about 10³. The parameter which most strongly affects the final mass ratio of the pair is the time at which the secondary embryo is introduced. Our model yields the proper Moon-Earth mass ratio if the Moon embryo is introduced when the Earth is only about $\text{1}\text{10}$ of its final mass. The lunar orbit remains at about 10 Earth radii throughout most of the growth.This model of satellite formation overcomes two difficulties of the “circumterrestrial cloud” model of Ruskol (1960, 1963, 1972): (1) The difficulty of accumulating a mass as great as the entire Moon before gravitational instability reduces the cloud to a small number of moonlets is removed. (2) The differences between terrestrial and outer planet satellite systems is easily understood in terms of the differences in Q between these planets. The high Q of the outer planets does not allow a satellite embryo to survive a significant portion of the accretion process, thus only small bodies which formed very late in the accumulation of the planet remain as satellites. The low Q of the terrestrial planets allows satellite embryos of these planets to survive during accretion, thus massive satellites such as the Earth's Moon are expected. The present lack of such satellites of the other terrestrial planets may be the result of tidal evolution, either infall following primary despinning (Burns, 1973) or escape due to increase in orbit eccentricity.     

Selenographic control

Evaluation of selenographic data obtained with use of different observational means require the formulation of rigorous algorithms connecting the systems of coordinates, which the various methods have been referred to. The lunar principal axes of inertia are suggested as most appropriate for reference in lunar mapping and selenographic coordinate catalogues. The connection between the instantaneous axis of lunar rotation (involved in laser ranging, radar studies, astronomical observations from the surface of the Moon and VLBI observations of ALSEPs), the ecliptic system of coordinates (which in reductions of observations was considered as fixed in space), the Cassini mean selenographic coordinates (to which physical libration measures were referred), the lunar principal axes of inertia and the invariable plane of the solar system is discussed.On leave from the University of Manchester, England.Lunar Science Institute Contribution No. 138.Communication presented at the Conference on Lunar Dynamics and Observational Coordinate Systems, Held January 15–17, 1973, at the Lunar Science Institute, Houston, Tex., U.S.A.     

月球卫星轨道力学综述

月球探测器的运动通常可分为3个阶段，这3个阶段分别对应3种不同类型的轨道：近地停泊轨道、向月飞行的过渡轨道与环月飞行的月球卫星轨道。近地停泊轨道实为一种地球卫星轨道；过渡轨道则涉及不同的过渡方式(大推力或小推力等)；环月飞行的月球卫星轨道则与地球卫星轨道有很多不同之处，它决不是地球卫星轨道的简单克隆。针对这一点，全面阐述月球卫星的轨道力学问题，特别是环月飞行中的一些热点问题，如轨道摄动解的构造、近月点高度的下降及其涉及的卫星轨道寿命、各种特殊卫星(如太阳同步卫星和冻结轨道卫星等)的轨道特征、月球卫星定轨等。     

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).     

Some effects of elasticity on lunar rotation

A general Hamiltonian for a rotating Moon in the field of the Earth is expanded in terms of parameters orienting the spin angular momentum relative to the pricipal axes of the Moon and relative to coordinate axes fixed in the orbital plane. The effects of elastic distortion are included as modifications of the moment of inertia tensor, where the magnitude of the distortion is parameterized by the Love numberk ₂. The principal periodic terms in the longitude of a point on the Moon due to variations of the tide caused by the Earth are shown to have amplitudes between 3.9 × 10^–3 and 1.6 × 10^–2 with a period of an anomalistic month, 3.0 × 10^–4 and 1.2 × 10^–3 with a period of one-half an anomalistic month and 2.4 × 10^–4 and 9.6 × 10^–4 with a period of one-half of a nodical month. The extremes in the amplitudes correspond to rigidities of 8 × 10¹¹ cgs and 2 × 10¹¹ cgs, respectively, the former rigidity being comparable to that of the Earth. Only the largest amplitude given above is comparable to that detectable by the projected precision of the laser ranging to the lunar retrorereflectors, and this amplitude corresponds to an improbably low rigidity for the Moon. A detailed derivation of the free wobble of the lunar spin axis about the axis of maximum moment of inertia is given, where it is shown that elasticity can alter the period of the free wobble of 75.3 yr by only 3 × 10^–4 to 10^–3 of this period. Also, the effect of elasticity on the period of free libration is completely negligible by many orders of magnitude. If the Moon's rigidity is close to that of the Earth there is no effect of elasticity on the rotation which can be measured with the laser ranging and, therefore, no elastic properties of the Moon can be determined from variations in the rotation.Currently on leave from the Dept. of Physics, University of California, Santa, Barbara, California.Communication presented at the conference on Lunar Dynamics and Observational Coordinate Systems held January 15–17, 1973 at the Lunar Science Institute, Houston, Tex., U.S.A.     

An analytic method to account for drag in the Vinti satellite theory

In order to retain separability in the Vinti theory of Earth satellite motion when a nonconservative force such as air drag is considered, a set of variational equations for the orbital elements are introduced, and expressed as functions of the transverse, radial, and normal components of the nonconservative forces acting on the system. In this approach, the Hamiltonian is preserved in form, and remains the total energy, but the initial or boundary conditions and hence the Jacobi constants of the motion advance with time through the variational equations. In particular, the atmospheric density profile is written as a fitted exponential function of the eccentric anomaly, which adheres to tabular data at all, altitudes and simultaneously reduces the variational equations to definite integrals with closed form evaluations whose limits are in terms of the eccentric anomaly. The values of the limits for any arbitrary time interval are obtained from the Vinti program.Results of this technique for the case of the intense air drag satellites San Marco-2 and Air Force Cannonball are given. These results indicate that the satellite ephemerides produced by this theory in conjunction with the Vinti program are of very high accuracy. In addition, since the program is entirely analytic, several months of ephemerides can be obtained within a few seconds of computer time.     

The brightness distribution over the moon's disk at 6 cm wavelength

On November 27, 1974, a map of the Moon was obtained at 6 cm wavelength with the 100-m-telescope in Effelsberg. The high angular and favourable temperature resolution allowed an interpretation of the observed brightness distribution. The dominant feature of the brightness distribution is the centre-to-limb variation, particularly noticeable in the direction of the poles. The exponent of the commonly adopted cos ()-law, describing the temperature variation across the lunar disk, is determined as 0.4. The North-South variation of the lunar surface temperature is estimated to be 30%; the depth of penetration (L _e) of electromagnetic waves of 6 cm wavelength is found to beL _e 17 m.     

The fate of primordial lunar Trojans

Multiple large impact basins on the lunar nearside formed in a relatively-short interval around 3.8-3.9 Gyr ago, in what is known as the Lunar Cataclysm (LC; also known as Late Heavy Bombardment). It is widely thought that this impact bombardment has affected the whole Solar System or at least all the inner planets. But with non-lunar evidence for the cataclysm being relatively weak, a geocentric cause of the Lunar Cataclysm cannot yet be completely ruled out [Ryder, G., 1990. Eos 71, 313, 322-323]. In principle, late destabilization of an additional Earth satellite could result in its tidal disruption during a close lunar encounter (cf. [Asphaug, E., Agnor, C.B., Williams, Q., 2006. Nature 439, 155-160]). If the lost satellite had D>500 km, the resulting debris can form multiple impact basins in a relatively short time, possibly explaining the LC. Canup et al. [Canup, R.M., Levison, H.F., Stewart, G.R., 1999. Astron. J. 117, 603-620] have shown that any additional satellites of Earth formed together with (and external to) the Moon would be unable to survive the rapid initial tidally-driven expansion of lunar orbit. Here we explore the fate of objects trapped in the lunar Trojan points, and find that small lunar Trojans can survive the Moon's orbital evolution until they and the Moon reach 38 Earth radii, at which point they are destabilized by a strong solar resonance. However, the dynamics of Trojans containing enough mass to cause the LC (diameters >150 km) is more complex; we find that such objects do not survive the passage through a weaker solar resonance at 27 Earth radii. This distance was very likely reached by the Moon long before the LC, which seems to rule out the disruption of lunar Trojans as a cause of the LC.     

Some orbital characteristics of lunar artificial satellites

In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.     

Theoretical line profiles from a symbiotic system

A model for a symbiotic system is presented which is not spherically symmetrical. The system consists of a cool, mass losing giant and a hot star. The stellar wind from the cool star is radiatively ionized by the hot companion, and the resulting nebular lines are emitted mainly close to the cool star. Due to the elliptical orbit, the line fluxes change with time because of the varying separation of the stars. The line profiles vary from symmetrical to asymmetrical, depending on the angle between the line of sight and the axis of symmetry. The calculated line profiles are compared with observed UV lines of the symbiotic star HBV 475.Paper presented at the IAU Colloquium No. 93 on Cataclysmic Variables, Recent Multi-Frequency Observations and Theoretical Developments, held at Dr. Remeis-Sternwarte Bamberg, F.R.G., 16–19 June, 1986.