A note on the physical parameters of the Moon

Using the Stokes coefficients of the Moon recommended by IERS Standards (1992), we determined the expression of lunar equipotential surface (selenoid), then calculated the parameters of best-fitting lunar ellipsoid. Moreover, we derived a set of hydrostatic values of lunar physical parameters by solving the Clairaut equation, and discussed the feature of nonhydrostatic component in the figure parameters of the Moon. Finally, we suggest that the lunar physical parameters should be divided into three kinds: primary constants, derived constants, and estimated constants (see Table III); as well that the second-degree Love number,k ₂, of the Moon should belong to the estimated constants. At present, in view of the accuracy in reductions of LLR data, a value ofk ₂ obtained from model calculations should be defined as its adopted value. For example, the value ofk ₂, 0.0266, in this paper can replace 0.0222 in IERS Standards (1992).     

A Huge Basin in the Unknown Portion of Mercury in the 250°–290° W Longitude Range

A basin with an interior about 1000 km across and an outer rim about 2000 km in diameter has been found on the side of Mercury that remained unknown after theMariner 10imaging in 1973–1975. In its size, which is as much as 1/2 of the Mercury's diameter, this formation is one of the largest basins on the terrestrial planets. The presence of other large features of global dimension on this side of the planet suggests their possible asymmetric allocation on Mercury, similar to that on the Moon. New methods of ground-based astronomical observations including data processing with a special software package have been used.     

On the internal structures of mercury and venus

Recent radar measures of the radius and mass of Mercury imply a composition for the planet containing about 60% iron. One or other of two conclusions seems inescapable: either that Mercury is a highly exceptional object among terrestrial planets, or that all measures to date of the planet involve substantial systematic error. In either case the situation is such that independent checking of the radius and mass of Mercury by some entirely different means has become of the greatest importance to planetary physics and cosmogony.The recent radar and other determinations of the solid radius of Venus imply an internal structure similar to that of the Earth, namely a liquid core surrounded by a solid mantle and outer-shell zone. The theory also implies that the temperatures within Venus should be slightly higher than at the corresponding parts of the Earth. The proportion of mass in the core of Venus (about 25% of the whole) is entirely consistent with the phase-change hypothesis as to its nature, as of course is also the absence of any liquid or iron core in both Mars and the Moon. On the older iron-core hypothesis, Venus with considerably less iron content by mass than the Earth, and Mars and the Moon with none, would all present problems in different degrees to account for the differences of composition.If Venus began as an all-solid planet, the initial radius would have been about 6300 km, and the total amount of surface reduction to date owing to contraction of the planet would have been almost 40 million km², and as a proportion of the total area only slightly less than the contraction of the Earth. The theory thus predicts the existence of folded and thrusted mountain-systems of terrestrial type at the surface of Venus.     

Thermal evolutions of the terrestrial planets

The thermal evolution of the Moon, Mercury, Mars, Venus and hypothetical minor planets is calculated theoretically, taking into account conduction, solid-state convection, and differentiation. An assortment of geological, geochemical, and geophysical data is used to constrain both the present day temperatures and thermal histories of the planets' interiors. Such data imply that the planets were heated during or shortly after formation and that all the terrestrial planets started their differentiations early in their history. Initial temperatures and core formation play the most important roles in the early differentiation. The size of the planet is the primary factor in determining its present day thermal state. A planetary body with radius less than 1000 km is unlikely to reach melting given heat source concentrations similar to terrestrial values and in the absence of intensive early heating such as short half-life radioactive heating and inductive heating.Studies of individual planets are constrained by varying amounts of data. Most data exist for the Earth and Moon. The Moon is a differentiated body with a crust, a thick solid mantle and an interior region which may be partially molten. It is presently cooling rapidly and is relatively inactive tectonically.Mercury most likely has a large core. Thermal calculations indicate it may have a 500 km thick solid lithosphere, and the core may be partially molten if it contains some heat sources. If this is not the case, the planet's interior temperatures are everywhere below the melting curve for iron. The thermal evolution is dominated by core separation and the high conductivity of iron which makes up the bulk of Mercury.Mars, intermediate in size among the terrestrial planets, is assumed to have differentiated an Fe–FeS core. Differentiation and formation of an early crust is evident from Mariner and Viking observations. Theoretical models suggest that melting and differentiation of the mantle silicates has occurred at least up until 1 billion years ago. Present day temperature profiles indicate a relatively thick (250 km) lithosphere with a possible asthenosphere below. The core is molten.Venus is characterized as a planet similar to the Earth in many respects. Core formation probably occurred during the first billion years after the formation. Present day temperatures indicate a partially molten upper mantle overlain by a 100 km thick lithosphere and a molten Fe–Ni core. If temperature models are good indicators, we can expect that today, Venus has tectonic processes similar to the Earth's.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May 1978.     

Secular love numbers and hydrostatic equilibrium of planets

The secular Love numbers of planets and of the Moon have been computed by use of the recent satellite data. It has been demonstrated that Venus, Mercury, and the Moon are far from the ideal hydrostatic equilibrium; the angular velocities (rotation periods) which correspond to the hydrostatic equilibrium have been computed.     

Could giant impacts cripple core dynamos of small terrestrial planets?

Large impacts not only create giant basins on terrestrial planets but also heat their interior by shock waves. We investigate the impacts that have created the largest basins existing on the planets: Utopia on Mars, Caloris on Mercury, Aitken on Moon, all formed at ∼4 Ga. We determine the impact-induced temperature increases in the interior of a planet using the “foundering” shock heating model of Watters et al. (Watters, W.A., Zuber, M.T., Hager, B.H. [2009]. J. Geophys. Res. 114, E02001. doi:10.1029/2007JE002964). The post-impact thermal evolution of the planet is investigated using 2D axi-symmetric convection in a spherical shell of temperature-dependent viscosity and thermal conductivity, and pressure-dependent thermal expansion. The impact heating creates a superheated giant plume in the upper mantle which ascends rapidly and develops a strong convection in the mantle of the sub-impact hemisphere. The upwelling of the plume rapidly sweeps up the impact-heated base of the mantle away from the core-mantle boundary and replaces it with the colder surrounding material, thus reducing the effects of the impact-heated base of the mantle on the heat flux out of core. However, direct shock heating of the core stratifies the core, suppresses the pre-existing thermal convection, and cripples a pre-existing thermally-driven core dynamo. It takes about 17, 4, and 5 Myr for the stratified cores of Mars, Mercury, and Moon to exhaust impact heat and resume global convection, possibly regenerating core dynamos.     

Resolved images of an unknown sector on the surface of Mercury

We took electronic photographs of Mercury on the side of the planet that was not photographed from the Mariner-10 spacecraft in 1973–1975 by the millisecond-exposure method in ground-based observations. Based on these photographs, we synthesized resolved images of the surface of unknown regions of the planet. The capabilities of the method are limited by the small angular size of the planetary disk (only 7.3 arcsec at average quadrature), specific difficulties of Mercury’s ground-based observations, their very limited duration, and the laboriousness of the subsequent computer-aided observational data processing. The millisecond-exposure method is complex, but a sufficient number of primary electronic photographs can be taken under good seeing conditions for the subsequent synthesis of Mercurian images with a resolution of no worse than the diffraction limit. A giant basin about 2000 km in diameter and other large structures are distinguished in the synthesized images of the planet. In the regions where radar data are available, these structures can be identified with previously found ones. In some measure, the synthesized images allow the relief of the longitude sector 210°–290° W to be reconstructed on Mercury. It can be asserted with caution that the large relief features are distributed asymmetrically over the surface of Mercury, much as observed on other terrestrial planets, the Moon, and many satellites of giant planets.     

Mercury: Internal structure and thermal evolution

Astronomical observations and cosmochemical calculations suggest that the planet Mercury may be composed of materials which condensed at relatively high temperatures in the primitive solar nebula and may have a basaltic crust similar to parts of the moon. These findings, plus the long standing inference that Mercury is much richer in metallic iron than the other terrestrial planets, provide important constraints which we apply to models of the thermal evolution and density structure of the planet. The thermal history calculations include explicitly the differing thermal properties of iron and silicates and account for core segregation, melting and differentiation of heat sources, and simulated convection during melting. If the U and Th abundances of Mercury are taken from the cosmochemical model of Lewis, then the planet would have fully differentiated a metal core from the silicate mantle for all likely initial temperature distributions and heat transfer properties. Density distributions for the planet are calculated from the mean density and estimates of the present-day temperature. For the fully differentiated model, the moment of inertia C/MR² is 0.325 (J₂=0.302×10^?6). For models with lower heat source abundances, the planet may not yet have differentiated. The density profiles for such models give C/MR²=0.394 (J₂=0.487×10^?6). These results should be useful for preliminary interpretation of the Mariner 10 measurements of Mercury's gravitational field.     

On planetary cores

The early part of the article traces the implications on planetary cores of various revisions of the writer's original Earth modelsA andB. Recent evidence indicates that it is appropriate to consider at most two-layer cores in the terrestrial planets. Current evidence on equations of state for cores is presented. The later part of the article puts forward a proposal which, though quite distinct from the mantle-core phase transition proposal of some twenty years ago, retains the assumption that the pressurepc at the Earth's mantle-core boundaryN is a critical pressure crucially involved in the changes atN. The proposal envisages an outer core of Fe₂O which is unstable at pressures less thanpc. In the form set down, the proposal restores the agreement of the Earth, Mars and Venus with a common overall composition and meets various revisions of earlier planetary and other observational data. Although the calculations so far made are only preliminary, the agreement appears to be very striking. While still requiring the present Mercury to be significantly different from the primitive Mercury, the Fe₂O has an advantage over the phase transition proposal in providing for a larger iron core. Brief comments are made on the Moon.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

The effects of nebula surface density profile and giant‐planet eccentricities on planetary accretion in the inner solar system

Abstract— We describe results of 32 N‐body planetary accretion simulations that investigate the dependence of terrestrial‐planet formation on nebula surface density profile σ and evolution of the eccentricities of Jupiter and Saturn e_j,s. Two surface density profiles are examined: a decaying profile with σ ∝ 1/a, where a is orbital semi‐major axis, and a peaked profile in which σ increases for a < 2 AU and decreases for a > 2 AU. The peaked profiles are generated by models of coagulation in an initially hot nebula. Models with initial e_j,s = 0.05 (the current value) and 0.1 are considered. Simulations using the decaying profile with e_j,s = 0.1 produce systems most like the observed planets in terms of mass‐weighted mean a and the absence of a planet in the asteroid belt. Simulations with doubled σ produce planets roughly twice as massive as the nominal case. Most initial embryos are removed in each simulation via ejection from the solar system or collision with the Sun. The asteroid belt is almost entirely cleared on a timescale of 10–100 Ma that depends sensitively on e_j,s. Most initial mass with a < 2 AU survives, with the degree of mass loss increasing with a. Mass loss from the terrestrial region occurs on a timescale that is long compared to the mass loss time for the asteroid belt. Substantial radial mixing of material occurs in all simulations, but is greater in simulations with initital e_j,s = 0.05. The degree of mixing is equivalent to a feeding zone of half width 1.5 and 0.9 AU for an Earth mass planet at 1 AU for the cases e_j,s = 0.05 and 0.1, respectively. In simulations with e_j,s = 0.05, roughly one‐third and 5–10% of the mass contained in final terrestrial planets originated in the region a > 2.5 AU for the decaying and peaked profiles, respectively. In the case e_j,s = 0.1, the median mass accreted from a > 2.5 AU is zero for both profiles.     

A comparison between two internal structure models of the Earth

In view of the elastic deformation of the Earth we performed the comparative study for the Earth's models 1066 A and PREM, calculated the static Love numbers from degree 2 to 30, and discussed the relative variations of the second degree Love numbers and their combinations due to the variation of the position of the core-mantle boundary, due to the redistribution of V _p, V _s, and in the lower mantle, and due to the possible rigidity in the outer core. From the above-mentioned discussions we recommended that the Standard Earth Model (SEM) should include two kinds of models—one is oceanless, and another has an oceanic surface. Finally, we calculated the astronomicgeodetic parameters, which are consistent with the primary constants in the IERS Standards, of the SEM.     

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.     

Simultaneous determination of global topography, tidal Love number and libration amplitude of Mercury by laser altimetry

Solar tidal forces generate elevation changes of Mercury's surface of the order 1 m within one Hermean year, and solar torques on the non-symmetric permanent mass distribution of the planet cause an uneven rotation of Mercury's surface with a libration amplitude of the order of 40 arcsec. Knowledge of the precise reaction of the planet to tidal forcing, expressed by the Love numbers h₂ and k₂, as well as accurate knowledge of the amplitude of forced libration Φ_lib, puts constraints on the internal structure, for example the state and the size of the core. The MESSENGER and BepiColombo missions to Mercury carry laser altimeters, whose primary goal is to accurately map the topography. Here we investigate if the Love number h₂ and the amplitude of forced libration can be determined together with the static topography of the planet from a global altimetry record. We do this by creating synthetic altimeter data for the nominal orbit of BepiColombo over the nominal mission duration of approximately four Mercury years and inverting them for the static and time-dependent parts of the topography. We assume purely Gaussian noise. We find that it is possible to extract both parameters h₂ and Φ_lib with an accuracy of approximately 10%, while the static topography coefficients of a spherical harmonic expansion can be determined simultaneously with an accuracy at the centimetre level. Extraction of the static topography to higher harmonic degrees improves the precision of the measurement of h₂ and Φ_lib. The simulation results demonstrate that it seems feasible to test current models on Mercury's interior with sufficient precision using BepiColombo Laser Altimeter data.     

The effect of the internal structure of Mars on its seasonal loading deformations

Mars is continuously subjected to surface loading induced by seasonal mass changes in the atmosphere and ice caps due to the CO₂ sublimation and condensation process. It results in surface deformations and in time variations of gravity. Large wavelength annual and semi-annual variations of gravity (particularly zonal coefficients ΔJn) have been determined using present day geodetic satellite measurements. However loading deformations have been poorly studied for a planet like Mars. In this paper, we compute these deformations and their effect on spacecraft orbiting around Mars. Loading deformations of terrestrial planet are typically investigated assuming a spherical planet, radially symmetric. The mean radial structure of Mars is not well known. In particular the radius of the liquid or solid core remains not precisely determined. One may then wonder what is the effect of these uncertainties on loading deformations. Moreover, Mars presents a strong topography and probably large lateral variations of crustal thickness (relative to the Earth). The paper answer the questions of what is the effect of such lateral heterogeneities on surface deformations, and is the classical way to calculate loading deformation well adapted for a planet like Mars. In order to answer these questions we have investigated theoretically loading deformations of Mars-like planets. We first investigated classical load Love numbers. We show that for degrees inferior to 10, the load Love numbers mainly depend on the radius of the core and on its state, and that for degree greater than 10, they depend on the mean radius of mantle-crust interface. Using a General Circulation Model (GCM) of atmosphere and ice caps dynamics we show that loading vertical displacements have a 4-5 cm magnitude and present a North-South pattern with periodic transitions. Finally we investigated the effect of lateral variations of the crustal thickness on these loading deformations. We show that thickness heterogeneities perturb the deformations and the time variation of gravity at about 0.5%. However this perturbation on ΔJn is only about 1‰ due to main direct attraction of surface fluid layers. We conclude that lateral variations of crustal thickness are today negligible. However, observation of load Love numbers would bring information on the radial internal structure of the planet, particularly on the core radius. ΔJn study would permit to infer the load Love number , particularly for degree 2 and 3, knowing surface fluid layer dynamics. However load Love numbers are quite small (about 0.05), and despite the present good agreement between GCM and ΔJn observations, will only be estimated in the near future when a slightly better precision in observation and modeling will make it possible to infer these numbers. The investigation of load Love number , which are larger than numbers, would be particularly interesting. It would permit to study degree 1 contribution of atmosphere and ice caps dynamics, which is the most important component of surface fluid dynamics on Mars. Surface displacement measurements would be necessary on a few places near the pole regions, which may be possible in the future, with a project involving precise positioning of a lander on the surface of Mars.     

Planetary core size: A seismological approach

A key parameter for understanding the geodynamics of a terrestrial planet is the size of its core. Numerical evaluation of 28 different interior structure models of Mercury, Venus, Earth, the Moon, and Mars suggests that there is an almost linear relationship between the core radius and the extent of the seismic P-wave core shadow. A scaling law is derived from a simple mantle density and velocity model that permits the interpretation of respective seismic measurements on terrestrial planetary bodies.     

A Postulated Planetary Collision, the Terrestrial Planets, the Moon and Smaller Solar-System Bodies

In a scenario produced by the Capture Theory of planetary formation, a collision between erstwhile solar-system giant planets, of masses 798.75 and 598.37 M _⊕, is simulated using smoothed-particle hydrodynamics. Due to grain-surface chemistry that takes place in star-forming clouds, molecular species containing hydrogen, with a high D/H ratio taken as 0.01, form a layer around each planetary core. Temperatures generated by the collision initiate D–D reactions in these layers that, in their turn, trigger a reaction chain involving heavier elements. The nuclear explosion shatters and disperses both planets, leaving iron-plus-silicate stable residues identified as a proto-Venus and proto-Earth. A satellite of one of the colliding planets, captured or retained by the proto-Earth core, gave the Moon; two massive satellites released into heliocentric orbits became Mercury and Mars. For the Moon and Mars, abrasion of their surfaces exposed to collision debris results in hemispherical asymmetry. Mercury, having lost a large part of its mantle due to massive abrasion, reformed to give the present high-density body. Debris from the collision gave rise to asteroids and comets, much of the latter forming an inner reservoir stretching outwards from the inner Kuiper Belt that replenishes the Oort Cloud when it is depleted by a severe perturbation. Other features resulting from the outcome of the planetary collision are the relationship of Pluto and Triton to Neptune, the presence of dwarf planets and light-atom isotopic anomalies in meteorites.     

Exoplanet's Figure and Its Interior

Along with the development of the observing technology, the observation and study on the exoplanets’ oblateness and apsidal precession have achieved significant progress. The oblateness of an exoplanet is determined by its interior density profile and rotation period. Between its Love number k₂ and core size exists obviously a negative correlation. So oblateness and k₂ can well constrain its interior structure. Starting from the Lane-Emden equation, the planet models based on different polytropic indices are built. Then the flattening factors are obtained by solving the Wavre's integro-differential equation. The result shows that the smaller the polytropic index, the faster the rotation, and the larger the oblateness. We have selected 469 exoplanets, which have simultaneously the observed or estimated values of radius, mass, and orbit period from the NASA (National Aeronautics and Space Administration) Exoplanet Archive, and calculated their flattening factors under the two assumptions: tidal locking and fixed rotation period of 10.55 hours. The result shows that the flattening factors are too small to be detected under the tidal locking assumption, and that 28% of exoplanets have the flattening factors larger than 0.1 under the fixed rotation period of 10.55 hours. The Love numbers under the different polytropic models are solved by the Zharkov's approach, and the relation between k₂ and core size is discussed.     

Construction of Martian Interior Model

We present the results of extensive numerical modeling of the Martian interior. Yoder et al. in 2003 reported a mean moment of inertia of Mars that was somewhat smaller than the previously used value and the Love number k₂ obtained from observations of solar tides on Mars. These values of k₂ and the mean moment of inertia impose a strong new constraint on the model of the planet. The models of the Martian interior are elastic, while k₂ contains both elastic and inelastic components. We thoroughly examined the problem of partitioning the Love number k₂ into elastic and inelastic components. The information necessary to construct models of the planet (observation data, choice of a chemical model, and the cosmogonic aspect of the problem) are discussed in the introduction. The model of the planet comprises four submodels—a model of the outer porous layer, a model of the consolidated crust, a model of the silicate mantle, and a core model. We estimated the possible content of hydrogen in the core of Mars. The following parameters were varied while constructing the models: the ferric number of the mantle (Fe#) and the sulfur and hydrogen content in the core. We used experimental data concerning the pressure and temperature dependence of elastic properties of minerals and the information about the behavior of Fe(γ-Fe ), FeS, FeH, and their mixtures at high P and T. The model density, pressure, temperature, and compressional and shear velocities are given as functions of the planetary radius. The trial model M13 has the following parameters: Fe#=0.20; 14 wt % of sulfur in the core; 50 mol % of hydrogen in the core; the core mass is 20.9 wt %; the core radius is 1699 km; the pressure at the mantle-core boundary is 20.4 GPa; the crust thickness is 50 km; Fe is 25.6 wt %; the Fe/Si weight ratio is 1.58, and there is no perovskite layer. The model gives a radius of the Martian core within 1600–1820 km while ≥30 mol % of hydrogen is incorporated into the core. When the inelasticity of the Martian interior is taken into account, the Love number k₂ increases by several thousandths; therefore, the model radius of the planetary core increases as well. The prognostic value of the Chandler period of Mars is 199.5 days, including one day due to inelasticity. Finally, we calculated parameters of the equilibrium figure of Mars for the M13 model: J ₂ ⁰ = 1.82 × 10^?3, J ₄ ⁰ = ?7.79 × 10^?6, e _c-m ^D = 1/242.3 (the dynamical flattening of the core-mantle boundary).     

On the determination of the long period tidal perturbations in the elements of artificial earth satellites

In the present article we develop the theory of the long period tidal effects in the motion of artificial satellites assuming the variability of elastic parameters of the Earth (Love numbers) across the parallels. The dependence of Love numbers on the longitude produces perturbations of the period of one day or less and hence is neglected in the present theory. In this respect we follow in the footsteps of Kaula (1969). If the deviations ofk ₂ andk ₃ from pure constants are not taken into consideration, then the perturbations caused by the variability ofk ₂ andk ₃ across the parallels will be misinterpreted as the perturbations caused byk ₄...-terms, and the spurious values ofk ₄... will be deduced. It is extremely doubtful, however, that the real effects caused byk ₄,k ₅,..., are significant enough to be detected. The short period effects with the period of the revolution of the satellite, or less, were removed from the differential equations for the variation of elements of the satellite by the averaging over the orbit of the satellite. These differential equations are in the form convenient for numerical integration over a long interval of time and also suitable for developing the tidal effects into trigonometric series with the arguments ω, Ω of the satellite andl, l′, F, D, Γ of the Moon. The numerical integration can be performed using some simple quadrature formula, without resorting to a predictor-corrector system.     

Cumulative subject index volumes 27–29

It is premature to establish a chronology for Mars and Mercury, relative to the known lunar chronology, to better than an order of magnitude. Lunar evidence neither requires nor excludes a “cataclysmic” episode of bombardment about 4.0 b.y. ago. Such a cataclysm might have resulted naturally from tidal disruption by a planet or collisional fragmentation in the asteroid belt of either a Uranus/Neptune-scattered planetesimal or a large asteroid, in which case any lunar cataclysm would have occurred as well on other planets. There is no independent evidence in Mariner 10 imagery for (or against) an early episodic bombardment on Mercury. Crater densities on plains units of the Moon, Mars, and Mercury have not been shown to be “strikingly similar” and do not imply, in the absence of definitive dynamical calculations of planetary impact rates of plausible populations of planetesimals, any similarity in the geological chronologies for those planets. Photogeological studies alone cannot determine absolute chronologies for planets. In combination with dynamical analyses, they can help us date to no better than a factor of 3 to 10 the formation of the Caloris Basin or the epoch when the Martian rivers ran.