Fossil figure contribution to the lunar figure

The unusual shape of the Moon given its present rotational and orbital state has been explained as due to a fossil figure preserving a record of remnant rotational and tidal deformation (Jeffreys, H. [1915]. Mem. R. Astron. Soc. 60, 187–217; Lambeck, K., Pullan, S. [1980]. Phys. Earth Planet. Interiors 22, 29–35; Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652–655). However, previous studies assume infinite rigidity and ignore deformation due to changes in the rotational and orbital potentials as the Moon evolves to the present state. We interpret the global lunar figure with a physical model that takes into account this deformation. Although the Moon deforms in response to rotational and orbital changes, a fossil figure capable of explaining the observed figure can be preserved by an elastic lithosphere.     

Numerical model of the Moon's rotation

We have integrated numercially the differential equtions for the Moon's rotation with respect to an inertial coordinate system, and the variational equations for (i) the six initial conditions of the rotation; (ii) the moment-of-inertia ratios and ; and (iii) the coefficients of the third-degree gravitational harmonics. When these integrations are used in conjunction with our current lunar-orbit and Earth-rotation models, and all of the relevant initial conditions and parameters are adjusted to fit five years of McDonald Observatory lunar laser ranging observations, the root-mean-square (rms) of the postfit range residuals is 28 cm. When we adjust the lunar-rotation initial conditions separately to fit the physical libration angles given by the numerical model of Williams (1975), we find an rms orientation difference over a six-year interval of 0.03 arcsecond, after removal of a constant bias. A similar comparison of our model with the semi-analytical model of Eckhardt (1981) yields an rms orientation difference of 0.2 arcsecond.     

Relativistic theory of the Moon's motion

Relativistic corrections for the elements and coordinates of the Moon have been obtained in the framework of the PPN-formalism. The influence of the coordinate conditions on the observational effects was studied.     

Perturbations of a close-earth satellite due to sunlight diffusely reflected from the Earth

A semianalytical method has been developed to calculate the radiation-pressure perturbations of a close-Earth satellite due to sunlight reflected from the Earth. It is assumed that the satellite is spherically symmetric and that the solar radiation is reflected from the Earth according to Lambert's Law. To account for the increasing reflectivity of the Earth toward the poles, its albedo is assumed to have a latitudinal dependence given bya=a ₀ +a ₂ sin². The effect of the terminator on the perturbations has been neglected. The perturbations within a particular revolution are given analytically, while the long-range perturbations are obtained by accumulation.     

Perturbations of a close-Earth satellite due to sunlight diffusely reflected from the Earth

A semianalytic method has been developed to calculate the radiation-pressure perturbations of a close-Earth satellite due to sunlight reflected from the Earth. The assumptions made are that the satellite is spherically symmetric and that the solar radiation is reflected from the Earth according to Lambert's Law with uniform albedo. By using expressions for the components of the radiation-pressure force due to Lochry, the expressions for the perturbations of the elements were developed into series in the true anomalyv. The perturbations within a given revolution can be obtained analytically by integrating with respect tov while holding all slowly varying quantities constant. The long-range perturbations are then obtained by accumulating the net perturbations at the end of each revolution.     

Direct perturbations of the planets on the Moon's motion

The aim of the present paper is to present the theoretical background of a method to compute the planetary perturbations on the Moon's motion. We formulate an algorithm based upon the Lie transform method and well-suited to the particular problem at hand.This algorithm is being implemented using Henrard's Semi-Analytical Lunar Ephemeris (SALE) as solution of the Main Problem and Bretagnon's planetary theory. The accuracy of the solution is intended to be about 0".001 for terms of period up to 2000 years.To illustrate the interest of our approach, we comment on some preliminary results obtained about the direct perturbations due to Venus on the Moon's longitude. The final results will be the subject of another paper.     

The jump of the second derivative of the Moon's elongation

We present some results of new calculations ofD(t)-the second derivative of the Moon's elongation as a function of time. The paper contains an explanation of the well-known R. Newton's effecthe rapid decline inD(t) from about 700 yr to about 1300 yr. The new graph ofD is based on the revised dates of the ancient eclipses and has a qualitatively different character; in particular, the decline inD(t) vanishes completely andD(t) oscillates at a roughly constant value, which coincides with the modern one. This fact agrees with the independent chronological results in the author's [7] paper.     

Evolution of NEO rotation rates due to close encounters with Earth and Venus

In this paper we study the statistical effect of planetary flybys on the rotation rates and states of Near Earth Objects (NEOs). Our approach combines numerical and analytical methods within a Monte Carlo model that simulates the evolution of the NEO spin rates. We take as input for the simulation a source distribution of spin states and evolve it to find their steady state distribution. In performing this evolution we track the changes in the spin rate and state distribution for the different components of the NEO population. We show that the cumulative effect of planetary encounters is to spin up the overall population of NEOs. This spin up effect holds on average only, and particular members of the population may experience an overall decrease in rotation rate. This effect is clearly seen across all components of the NEO population and is significant both statistically and physically. For initially slow rotators the spin up effect is strong, lowering the mean rotation period by 32%. For faster rotating populations the effect is less, lowering the spin period by 15% for the intermediate case, 6% for fast rotating rubble piles, and 8% for fast rotating monoliths. Physically, the spin up effect pushes 1% of the fast rotating rubble-pile NEOs over the disruption limit, while 6% of these bodies experience a sub-disruption event that could modify their physical structure. For monolithic NEOs, the spin up effect is self-limiting, reaching a minimum spin period of 1.1 hr, with a strong cut-off between 2-3 hr. This has two implications. First, it may not be necessary to invoke the rubble-pile hypothesis to recover a cut-off in spin period. Second, it shows that planetary flybys cannot account for the extremely rapid rotation rates of some small NEOs. We also tested a different balance between the effects of Earth and Venus by treating the Aten sub-class of asteroids separately. Due to increased interactions with the planets, the spin up effect is more pronounced (10%) and disruptions increase by a factor of three. The slow rotation tails of the spin distributions are increased to longer periods, in general, with rotation periods of over 100 hr occurring for a few tenths of a percent for some component populations. Thus, this mechanism may account for some of the noted excess in slow rotators among the NEOs. Planetary flybys also cause NEOs to enter a tumbling state, with approximately 0.5% of the population being placed into a long-axis rotation mode. Finally, based on the evolution of spin states of different components of the NEO population, we compared the evolved states with the measured distribution of NEOs to estimate the relative populations of these components that comprise the NEOs.     

Observations of the Solar Limb Shape Distortions

Two new campaigns devoted to the observation of the solar limb distortions were made at the Pic-du-Midi Observatory, in September 2000 and September 2001, by means of the scanning heliometer. This apparatus can be used now routinely to accurately determine solar limb profiles (at two wavelengths), at any heliographic latitudes. Each measurement is made within 44 milliseconds (of time) which permits to record a limb profile together with the seeing. Scans are automatically rejected for seeing larger than 1.3 arc sec. Such conditions are essential to perform high-quality observations necessary to obtain the quadrupole term (l=2) in the polynomial expansion of the radius contour R() _{= constant} = R ₀ left(1+_l c _l P _l()right). Exceptional meteorological conditions in September 2001 (seeing of the order of 18 cm, for a 50 cm clear aperture of the refractor) enabled us to determine c ₂ and c ₄ (see Table I) with an accuracy of a few milli-arc-sec. Results indicate a distorted solar shape, the departures from a pure spherical body not exceeding 20 milli-arc-sec. We propose a model to interpret such results (the combination of a nearly uniform rotating core with a prolate solar tachocline and an oblate surface), which is briefly discussed. Our results are confronted to those obtained from space. We conclude that measurements of the quadrupole term from the ground are possible, but of high difficulty and can be obtained only during excellent weather conditions. The hexadecapole term should be only obtained from space. We show that an astrometric satellite would be required, whose mission would be also to accurately determine the solar rotation profiles (both surface and in depth) in order to unambiguously determine the inertia moments of the Sun through the J _n terms. Such values are also briefly discussed.     

Numerical simulation of the Moon's rotation in a rigorous relativistic framework

This paper describes a numerical simulation of the rigid rotation of the Moon in a relativistic framework.Following a resolution passed by the International Astronomical Union(IAU) in 2000,we construct a kinematically non-rotating reference system named the Selenocentric Celestial Reference System(SCRS) and give the time transformation between the Selenocentric Coordinate Time(TCS) and Barycentric Coordinate Time(TCB).The post-Newtonian equations of the Moon's rotation are written in the SCRS,and they are integrated numerically.We calculate the correction to the rotation of the Moon due to total relativistic torque which includes post-Newtonian and gravitomagnetic torques as well as geodetic precession.We find two dominant periods associated with this correction:18.6 yr and 80.1 yr.In addition,the precession of the rotating axes caused by fourth-degree and fifth-degree harmonics of the Moon is also analyzed,and we have found that the main periods of this precession are 27.3 d,2.9 yr,18.6 yr and 80.1 yr.     

Lunar and terrestrial tidal effects on the Moon's rotational motion

An accurate model of the rotation of the Moon, constructed by numerical integration, has been presented in a previous paper. All direct perturbations capable of producing at least 10^–4 seconds of arc on the Moon's rotational motion have been included, and the physical librations resulting from planetary effects and Earth-Moon figure-figure interactions have been presented. The present study deals with the Moon's physical librations resulting from the non-rigidities of the Moon and the Earth. The effects of the Moon's elasticity and of a lunar phase lag are analyzed. Physical librations due to lunar tides and those due to terrestrial tides are presented and described.     

Direct perturbations of the planets on the Moon's motion: Results and comparisons

Celestial Mechanics and Dynamical Astronomy - The aim of this paper is to present the principal features of a new evaluation of the direct perturbations of the planets on the motion of the Moon....     

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.     

Comments about the direct perturbations of Venus and Mars on the Moon's motion

In a previous paper (Standaert, 1980) we have described an algorithm to compute the direct perturbation of the planets on the Moon's motion. A short summary of this algorithm is presented in Section 2 of this paper. Our first results permit us to present some complements and comments about these computations.The algorithm is based upon the Lie transform method and is implemented using Chapront's ELP as solution of the main problem with the partial derivatives of Henrard's Semi-Analytical Lunar Ephemeris (SALE), and Bretagnon's mean Keplerian orbit.An analysis of truncation errors in intermediate results is presented including the resonance effects. The final accuracy of the solution is intended to be about 0.0005 for terms of period up to 2000 yr in the case of Venus and up to 5000 yr in the case of Mars.The effects of second-order terms in the masses are investigated. Only those depending upon the second derivatives of the mean motions are found to be significant to the given accuracy and are included.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

On the figure of the planet Mercury

The figure of Mercury is estimated in terms of an isostatic form of equilibrium which tends to be controlled by the situation near perihelion passage at the 32 resonance spin rate. The ratios of the principal moments of inertia for Mercury are: (1)(C–A)/C7×10^–5; (2)(C–B)/C5×10^–5 and (3)(B–A)/C2×10^–5. The thermal effect on Mercury's figure during solidification forces Mercury's rotation to be trapped in the 32 resonance lock as its spin rate is being slowed by tidal effects. It is shown that the process of trapping of Mercury has been naturally affected by the instantaneous solidification of Mercury into a shape with two thermal bulges, and that the two permanent thermal bulges stabilize the planet's rotation.     

Possible changes of the AES orbit caused by a passage through the Moon's shadow

The change in the eccentricity of the AES orbit of the type 1963-30D as a result of one passage through the Moon's shadow is estimated, and it has been shown that the effect is not negligible.     

The representation of the force functions of the Earth's and Moon's gravitation in ellipsoidal coordinates

This paper derives asymptotic expansions of ellipsoidal coordinates in Cartesian coordinates and an expansion in spherical harmonics of the dominant term for the solution of Laplace's equation corresponding to the gravitational force function for a two-dimensional finite body.On comparing the expansion of the dominant term derived here with known expansions of the force functions of the Earth's and Moon's gravitation the author obtains values for the semimajor axes and eccentricities of the singular ellipses of these bodies in terms of the second degree harmonic coefficientsc ₂₀ andc ₂₂.     

Migration of Trans-Neptunian Objects to the Earth

Migration of trans-Neptunian objects under their mutual gravitation influence and the influence of the giant planets is investigated. These investigations are based on computer simulation results and on some formulas. We estimated that about 20 % of near-Earth objects with diameter d ≥ 1 km may have come from the Edgeworth-Kuiper belt. This revised version was published online in July 2006 with corrections to the Cover Date.