The evolution of the moon

The thermal evolution of the Moon as it can be defined by the available data and theoretical calculations is discussed. A wide assortment of geological, geochemical and geophysical data constrain both the present-day temperatures and the thermal history of the lunar interior. On the basis of these data, the Moon is characterized as a differentiated body with a crust, a 1000-km-thick solid mantle (lithosphere) and an interior region (core) which may be partially molten. The presence of a crust indicates extensive melting and differentiation early in the lunar history. The ages of lunar samples define the chronology of igneous activity on the lunar surface. This covers a time span of about 1.5 billion yr, from the origin to about 3.16 billion yr ago. Most theoretical models require extensive melting early in the lunar history, and the outward differentiation of radioactive heat sources.Thermal history calculations, whether based on conductive or convective computation codes define relatively narrow bounds for the present day temperatures in the lunar mantle. In the inner region of the 700 km radius, the temperature limits are wider and are between about 100 and 1600°C at the center of the Moon. This central region could have a partially or totally molten core.The lunar heat flow values (about 30 ergs/cm²s) restrict the present day average uranium abundance to 60 ± 15 ppb (averaged for the whole Moon) with typical ratios of K/U = 2000 and Th/U = 3.5. This is consistent with an achondritic bulk composition for the Moon.The Moon, because of its smaller size, evolved rapidly as compared to the Earth and Mars. The lunar interior is cooling everywhere at the present and the Moon is tectonically inactive while Mars could be and the Earth is definitely active.     

Thermal evolution of the moon

The thermal history and current state of the lunar interior are investigated using constraints imposed by recent geological and physical data. Theoretical temperature models are computed taking into account different initial conditions, heat sources, differentiation and simulated convection. To account for the early formation of the lunar highlands, the time duration of magmatism and presentday temperatures estimated from lunar electrical conductivity profiles, it is necessary to restrict initial temperatures and abundances of radioactivie elements. Successful models require that the outer half of the Moon initially heated to melting temperatures, probably due to rapid accretion. Differentiation of radioactive heat sources toward the lunar surface occurred during the first 1.6 billion years. Temperatures in the outer 500 km are currently low, while the deep interior (radius less than 700 to 1000 km) is warmer than 1000°C, and is of primordial material. In some models there is a partially melted core. The calculated surface heat flux is between 25 and 30 erg/cm² s.Presently at the Research Triangle Institute, Research Triangle, North Carolina 27709, U.S.A.     

Thermal history and evolution of the moon

Possible models for the thermal evolution of the Moon are constrained by a wide assortment of lunar data. In this work, theoretical lunar temperature models are computed taking into account different initial conditions to represent possible accretion models and various abundances of heat sources to correspond to different compositions. Differentiation and convection are simulated in the numerical computational scheme.Models of the thermal evolution of the Moon that fit the chronology of igneous activity on the lunar surface, the stress history of the lunar lithosphere implied by the presence of mascons, and the surface concentrations of radioactive elements, involve extensive differentiation early in lunar history. This differentiation may be the result of rapid accretion and large-scale melting or of primary chemical layering during accretion. Differences in present-day temperatures for these two possibilities are significant only in the inner 1000 km of the Moon and are not resolvable with presently available data.If the Apollo 15 heat flow is a representative value, the average uranium concentration in the moon is 65±15 ppb. This is consistent with achondritic bulk composition (between howardites and eucrites) for the Moon.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

The evolution of the moon: A finite element approach

Thermal convection has considerable influence on the thermal evolution of terrestrial planets. Previous numerical models of planetary convection have solved the system of partial differential equations by finite difference methods, or have approximated it by parametrized methods. We have evaluated the applicability of a finite element solution of these equations. Our model analyses the thermal history of a self-gravitating spherical planetary body; it includes the effects of viscous dissipation, internal melting, adiabatic gradient, core formation, variable viscosity, decay of radioactive nucleides, and a depth dependent initial temperature profile. Reflecting current interest, physical parameters corresponding to the Moon were selected for the model.Although no initial basalt ocean is assumed for the Moon, partial melting is observed very early in its history; this is presumably related to the formation of the basalt maria. The convection pattern appears to be dominated by an L-2 mode. The present-day lithospheric thickness in the model is 600 km, with core-mantle temperatures close to 1600 K. Surface heat flux is 25.3 mW m^–2, higher than the steady state-value by about 16%.The finite element method is clearly applicable to the problem of planetary evolution, but much faster solution algorithms will be necessary if a sufficient number of models are to be examined by this method.Notation coefficient of thermal expansion - _ij Kronecker delta - absolute or dynamic viscosity - perturbation in temperature - thermal diffusivity - kinematic viscosity - density - stress tensor - B.P. before present - c specific heat at constant pressure or volume (Boussinesq approximation) - d depth of convection - E ^* activation energy for creep - g gravity - Ga billions of years - H(t) heat generation per unit mass per unit time at timet - k Boltzmann's constant - K mean thermal conductivity - Ma millions of years - p pressure - q heat flux - q _ss steady-state heat flux - Ra Rayleigh number - S volumetric heat sources, includes radioactivity and viscous dissipation - t time - T temperature - u verocity vector - V ^* activation volume for creep     

Effect of a giant impact on the thermal evolution of the moon

The effects of a giant impact on the thermal evolution of the Moon are investigated. It is found that an impact similar to that of Imbrium creates lateral temperature variations of more than 200 deg within the upper 200 km of the Moon. Starting with a common lithosphere of 70 km thickness, the lithosphere beneath the basin grows to 200 km in thickness within 0.5 b.y. after the impact, while that beneath the highlands reaches to only 100 km in thickness during this period. The model presented for the thermal evolution of the Moon is compatible with (1) the existence of mascons for more than 3 b.y., (2) the late magmatization and subsequent volcanic activities during 4 to 3 b.y. ago and (3) the negative gravity rings around the large mascons.     

Electrical conductivity,internal temperatures and thermal evolution of the moon

The electrical conductivity of olivine and pyroxene is a strong function of the fugacity of oxygen in the atmosphere with which the mineral is in equilibrium. Lunar temperature profiles calculated from data on the electrical conductivity of these two minerals at oxygen fugacities similar to those which exist in the Moon indicate considerably higher temperatures for the lunar interior than obtained from conductivity data collected under normal atmospheric conditions. These high interior temperatures, the extensive differentiation associated with the formation of the lunar maria, and the radioactive element content of the Moon indicate that the Moon accreted at temperatures between 600 and 1000°C. Gravitational heating during accretion would lead to melting of at least the outer 200 km of the Moon and would produce conditions favourable to separation of a metal-sulfide melt sufficient to form a core of 200–300 km radius. Such a core would reach the center of the Moon within a few million years after accretion. This core could produce the remanent magnetization observed in the surface rocks. Dynamo action would cease with the cessation of convective motion within the core as the temperature of the surrounding mantle increased due to radioactive heating. With the radioactivity assumed in the present model and the high accretion temperature, this event would require less than 2 b.y., but more than 1.6 b.y.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

Viscosity of the moon

Using data from the present gravitational potential and surface topography of the Moon, it is possible to determine a lower limit of about 5 b.y. for the relaxation time of the mascons. Assuming that the Moon has behaved as a Maxwellian viscoelastic body since the formation of the mascons, this relaxation time indicates a value of about 10²⁷ poise for the viscosity of the lunar interior. Such a high viscosity implies that there has been no convection current inside the upper 800 km of the Moon since the formation of the mascons. Lunar Science Institute Contribution No. 99. The research in this paper was done while the author was a Visiting Scientist at the Lunar Science Institute, which is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration.     

Magnetism of the moon

The Malkus theory of a precessionally driven magnetoturbulence in a liquid core is applied to the Moon. It is shown that a lunar magnetic field requires the presence of a non-metallic core at at least 2500K or of an iron core at at least 2000K. Within the limits of our present knowledge these requirements may have been satisfied in the past. A new mechanism is proposed which is based on tidal effects in the outer solid and liquid shells whose existence is suggested by measurements of lunar radioactivity. This mechanism could account for the generation of local rather than poloidal fields at low latitudes in agreement with observation.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973.     

Conjectures on the ellipticity of cD galaxies

We give a possible explanation for the peculiar flattening of some cD galaxies, and for their alignment with the cluster's overall rotation.     

Accretion process of the moon

Recent geochemical and geophysical data suggest that the initial temperature of the Moon was strongly peaked toward the lunar surface. To explain such an initial temperature distribution, a simple model of accretion process of the Moon is presented. The model assumes that the Moon was formed from the accumulation of the solid particles or gases in the isolated, closed cloud. Two equations are derived to calculate the accretion rate and surface temperature of the accreting Moon. Numerical calculations are made for a wide range of the parameters particle concentration and particle velocity in the cloud. A limited set of the parameters gives the initial temperature profiles as required by geochemical and geophysical data. These models of the proto-moon cloud indicate that the lunar outershell, about 400 km thick, was partially or completely molten just after the accretion of the Moon and that the Moon should have been formed in a period shorter than 1000 yr. If the Moon formed at a position nearer to the Earth than its present one, the Moon might have been formed in a period of less than one year.On leave from Geophysical Institute, University of Tokyo.Contribution No. 2104, Division of Geological and Planetary Sciences, California Institute of Technology.     

The formation of the moon

Supporting evidence for the fission hypothesis for the origin of the Moon is offered. The maximum allowable amount of free iron now present in the Moon would not suffice to extract the siderophiles from the lunar silicates with the observed efficiency. Hence extraction must have been done with a larger amount of iron, as in the mantle of the Earth, of which the Moon was once a part, according to the fission hypothesis. The fission hypothesis gives a good resolution of the tektite paradox. Tektites are chemically much like products of the mantle of the Earth; but no physically possible way has been found to explain their production from the Earth itself. Perhaps they are a product of late, deep-seated lunar volcanism. If so, the Moon must have inside it some material with a strong resemblance to the Earth's mantle. Two dynamical objections to fission are shown to be surmountable under certain apparently plausible conditions.     

Topocentric aberration of the moon

Aberrational displacement of the observed topocentric positions of the Moon differ from the aberrational effect in its apparent ephemeris geocentric coordinates. The differential aberrational corrections due to the mutual positions of the observer and the Moon, may account to 0 _. ^″ 3. The reduction method of astrometric observations of the Moon, which takes into account this effect, is proposed.     

Pre-capture orbits of the moon

If the mass of the Earth was not considerably larger than at present, the pre-capture orbit of the Moon was in the range 0.9–1.1 A.U. Capture occurred within several 10⁸ years after formation of the Moon.     

Stress differences in the moon as an evidence for a cold moon

Assuming that the lateral variations of density in the lunar crust, the crustal density anomalies, are responsible for the lateral undulations of the lunar gravitational potential, we compute these anomalies for four different lunar models, which include an entirely solid Moon and three different solid lunar models with partially molten layers located within 600 km depth. The stress differences created by the density anomalies are determined for these models. It is found that, since the formation of the mascons, the entirely solid lunar model should have supported stress differences of the order of 70 bars while in the case of the other models, the solid layer overlying the partially molten one should have supported stress differences of more than 100 bars. The high stress differences associated with the partially molten models lead us to conclude that these models are not proper ones, and thus the Moon has always been solid since the formation of the mascons. Lunar Science Institute Contribution No. 97. The research in this paper was done while the author was a Visiting Scientist at the Lunar Science Institute, which is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration.     

Shock metamorphism on the moon

Shock metamorphism of the lunar samples is discussed. All types of lunar glasses formed by various-size collision-type impact are found as impact glass, ropy glass and agglutinates. The agglutinates bonded by crystal and glassy materials contain hydrogen and helium from the solar wind components. Lunar shocked minerals of plagioclase and silica show anomalous compositions and densities. There are typical two formation processes on planetary materials formed by shock events; that is (1) shocked quartz formed by silica-rich target rocks (esp. on evolved planets of the Earth and Mars), and (2) shocked silica with minor Al contents formed from plagioclase-rich primordial crusts of the Moon. The both shocked silica grows to coarse-grain normal crystals after high-temperature metamorphism which cannot distinguish the original main formation event of impact process.     

The thermal history of the moon

A simple analysis shows that the normal assumption of an outward heat flow, together with the normally assumed surface layer of low thermal conductivity, would give rise to microwave emission effects and to local variations in surface temperature which are not in fact observed. It is concluded that either the surface layer must be much thinner than is at present postulated, or that the outward flow of heat must be much smaller than is supposed.     

On periodic flybys of the moon

This paper considers the plane circular restricted three-body problem for small . Symmetric periodic solutions of the second species (passing near the body of mass ) and their distance from the center of the body of mass are studied by constructing perturbations of arc-solutions (solutions with consecutive collisions) existing for =0. Orbits which also pass near the body of mass 1- are studied in detail. The results are applied to finding periodic orbits in the Earth-Moon system and in the Sun-Jupiter system.Russian version: Preprint No. 91 (1978) of Inst. Appl. Math.; present English translation was made by L. M. Perko and W. C. Schulz (February 1979).