The provenance and evolution of comets

The process of comet formation through the hierarchical aggregation of originally submicron-sized interstellar grains to form micron-sized particles and then larger bodies in the protoplanetary disc, culminating in the formation of planetesimals in the disc extending from Jupiter to beyond Neptune, is briefly reviewed. The planetesimal theory for the origin of comets implies the existence of distinct cometary reservoirs, with implications for the immediate provenance of observed comets (both long-period and short-period) and their evolution as a result of planetary perturbations and physical decay, for example splitting and sublimation. The principal mode of cometary decay and collisional interaction with the terrestrial planets is through the formation and evolution of streams of cometary debris and hitherto undiscovered families of cometary asteroids. Recent dynamical results, in particular the sungrazing and sun-colliding end-state for short-period comet and asteroid orbits, are briefly discussed.     

Resonance rotation of celestial bodies and Cassini's laws

Celestial body rotation about its center of mass, taking into account the body orbit evolution, is considered. Non-linear evolution equations of motion are constructed. Empirical Cassini's laws describing the Moon motion result from these equations as their stationary points. Bifurcation conditions of steady motions are written out and conditions of their stability are investigated. Hypothesis of Mercury's resonance motion analogous to the motion by Cassini is discussed. Consequences of this hypothesis are considered.The first information including the results mentioned in the paper was previously published in preprint [1].     

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).     

Origin and evolution of the earth-moon system

The origin and evolution of the Earth-Moon system is studied by comparing it to the satellite systems of other planets. The normal structure of a system of secondary bodies orbiting around a central body depends essentially on the mass of the central body. The Earth with a mass intermediate between Uranus and Mars should have a normal satellite system that consists of about half a dozen satellites each with a mass of a fraction of a percent of the lunar mass. Hence, the Moon is not likely to have been generated in the environment of the Earth by a normal accretion process as is claimed by some authors.Capture of satellites is quite a common process as shown by the fact that there are six satellites in the solar system which, because they are retrograde, must have been captured. There is little doubt that the Moon is also a captured satellite, but its capture orbit and tidal evolution are still incompletely understood.The Earth and the Moon are likely to have been formed from planetesimals accreting in particle swarms in Kepler orbits (jet streams). This process leads to the formation of a cool lunar interior with an outer layer accreted at increasingly higher temperatures. The primeval Earth should similarly have formed with a cool inner core surrounded in this case by a very strongly heated outer core and with a mantle accreted slowly and with a low average temperature but with intense transient heating at each individual impact site.     

Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.     

On the stability of circular ‘asteroid’ orbits in anN-planetary system

The restricted problem of the motion of a point of negligible mass (asteroid) in anN-planetary system is considered. It is assumed that all the planets move about the central body (Sun) along circular orbits in the same plane and the mean motions of the asteroid and the planets are incommensurable. The asteroid orbit evolution is described as a first approximation by secular equations with the perturbing function averaged by the mean longitudes of the asteroid and the planets. For small values of the asteroid orbit eccentricity an expression for the secular part of the perturbing function has been obtained. This expression holds for the arbitrary values of the asteroid orbit semiaxis which are different from those of the planet orbit radii. The stability of the asteroid circular orbits in a linear approximation with respect to the eccentricity is studied. The critical inclinations for a Solar system model are calculated.     

On the Stability of the Terrestrial Planets as Models for Exosolar Planetary Systems

All results, achieved up to now, show the long term stability of our planetary system, although, especially the inner solar system is chaotic, due to some specific secular resonances. We study, by means of numerical integrations, the dynamical evolution of the planetary system where we concentrate on the stability of motion of the terrestrial planets Venus, Earth and Mars. Our model consists of a simplified planetary system with the inner planets Venus, Earth and Mars as well as Jupiter and Saturn. A mass factor was introduced to uniformly change the masses of the terrestrial planets; Jupiter and Saturn were involved in the system with their actual masses. We integrated the equations of motion with a Lie-integration method for a time interval of 10⁷ years. It turned out that when 220 < < 245 and > 250 the system became unstable due to the strong interactions between the planets. We discuss the model planetary systems for small mass-factors 0.5 10 and large ones 160 270 with the aid of several different numerical tools. These results can be applied to recently discovered exoplanetary systems, which configuration is comparable to our own.     

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.     

Lunar evolution: How well do we know it now?

The currently known astronomical, chemical and magnetic data are not uniquely indicative of an extensively and globally molten Moon. We argue here for an accretional layering in the Moon, but at temperatures below solidus. The excess mass in the near side of the Moon compatible with a 2 km displacement in the center of mass relative to the centre of figure and the moment of inertia data is considered to be due to Fe-FeS liquid formation and inhomogeneous segregation. These Fe-FeS bodies, termed fescons, are shown to be capable of accounting for the presently available magnetization data, by acting as small regenerative dynamos with a time-stability less than that of the terrestrial equivalent. The chemical characteristics of the highly differentiated materials (KREEP, granite etc.) are considered to be due to small scale localized melting caused by collisional events, from sources in which accessory phases play a significant role. Mare basalts are considered to be melts in the overlying material produced at a later time by⁴⁰K radioactivity in the fescons. Some consequences of the present hypothesis are suggested.We conclude that these and other characteristics of the lunar materials are reconcilable with a cold Moon such as discussed by Urey over the past two decades.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973.     

Tidal effects and the motion of a satellite

The effect of resonant planetary perturbations on the evolution of the orbit of a satellite driven by tidal forces is studied in this paper. The basic equations that govern it are similar to the equations found in orbit-orbit and in spin-orbit couplings. The general form of these equations is: A general treatment of such equations, proposed earlier (J. Kovalevsky, in Dynamical Trapping and Evolution of the Solar system, IAU Colloquium n^o74, V. V. Markellos and Y. Kozai, eds., 1983) is sketched.In particular, the effects of the large long periodic variations of the excentricity e' of the planet are analysed on an example taken from the lunar theory and the Earth's general theory due to Bretagnon.The argument of the well known planetary term =18 V-16T due to the tidal friction and quasi-periodic variations due to the presence of e' in the expression of the mean motion of the Moon. Their joint effect, has been to produce in the past resonant situations for this argument that repeated more than 100 times. Every such situation can be treated by equation (1).Numerical integration, using conditions that might have occurred while or similar other arguments were quasi resonant, have produced the following results: (a) In some cases, the argument becomes temporarily resonant. Between the capture to and the escape from the resonance, the semi-major axis undergoes oscillations, but the tidal secular evolution is stopped. (b) In other cases, the argument is not trapped into a resonant conditions, but the semi-major axis undergoes a quick change while d/dt is close to zero.A number of arguments that have been quasi resonant in the past history of the Earth-Moon system has been identified from the Chapront and Chapront-Touzé Lunar Theory. It appears that the phenomena described are frequent features in the evolution of the Lunar orbit.     

Perturbations by the oblateness of the Earth and by the planets in the motion of the Moon

Perturbations in the motion of the Moon are computed for the effect by the oblateness of the Earth and for the indirect effect of planets. Based on Delaunay's analytical solution of the main problem, the computations are performed by a method of Fourier series operation. The effect of the oblateness of the Earth is obtained to the second order, partly adopting an analytical evaluation. Both in longitude and latitude are found a few terms whose coefficient differs from the current lunar ephemeris based on Brown's theory by about 0.01. While, concerning the indirect effect of planets, several periodic terms in the current ephemeris seem to have errors reaching 0.05.As for the secular variations of and due to the figure of the Earth and the indirect effect of planets, the newly-computed values agree within 1/cy with Brown's results reduced to the same values of the parameters. Further, the accelerations in the mean longitude, and caused by the secular changes in the eccentricity of the Earth's orbite and in the obliquity of the ecliptic are obtained. The comparison with Brown shows an agreement within 0.3/cy² for the former cause and 0.02/cy² for the latter. An error is found in the argument of the principal term for the perturbations due to the ecliptic motion in the current ephemeris.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.     

Simple mass distribution for the lunar potential

A set of twenty-one point masses gravitationally equivalent to the L1 lunar potential model is presented. By construction, the equivalence is valid only in a region of space sampled by Apollo spacecraft. That region is taken to be a finite, torus-shaped shell. When used in place of the L1 model for Apollo 12 lunar orbit determination, the solution set gives spacecraft positions identical to within about 100 m.The solution is developed in two steps: first the L1 potential is examined to determine favorable mass locations, and then the mass values are computed to force an optimum matching of the L1 potential. Therefore the solution set is artificial. It is related to the Moon's actual mass distribution only in its similar gravitational effects in a limited region of space.     

Multiple expansion of the tidal potential

The Earth tidal deformation causes an additional gravitational potential. Its effect on the Moon orbital motion has been studied by several authors.In this contribution, we develop this additional potential without specifying the inertial frame chosen.For this purpose, we use the properties of the representation of rotation groups in 3 dimensions space. We finally obtain the interaction potential between the distorted Earth and the Moon which is a necessary preliminary to the study of the evolution of the Earth-Moon system.Nomenclature T.R.O Tide raising object - (, , ) Spherical coordinates of the T.R.O. - (J, _E ) Earth spin axis orientation. _E is the longitude of the ascending node of Earth's equator on thexy-plane - (a ,I ,e , , ,M ) Elliptics elements of the T.R.O     

On the motion of a satellite in resonance with its rotating planet

The influence of resonance perturbations due to the gravitational field of an oblate planet on its satellite whose motion is commensurable with rotation of the planet has been investigated. It has been shown that in special case of the critical inclination or circular orbit the Lagrange equations can be integrated for all resonance terms simultaneously. The method is applied to the investigation of the motion of the 12-hour communication and navigation satellites of the Molniya and Navstar type. The computations has been performed by the use of four models of the geopotential.     

Improvements of Planetary Theories Over 6000 Years

Taking advantage of the JPL Long Ephemeris DE406 and of the semi-analytical solution for the planetary motions VSOP87, we make an approximation of the differences JPL–VSOP. The form of the approximation (arguments and Poisson expansions) is analogous to the time series of the theory. We improve the planetary solution VSOP87 in two directions: a better fit of the integration constants via the highly precise observations used in DE406 and an extension of the length of validity covering the 6000 years of the source ephemeris. Over an interval of 2000 years, we achieve on the mean longitudes (test variables) a precision better than 0.005 for inner planets and 0.015 for outer ones. On the longest time interval covering 6000 years the precision is better than 0.03 for inner planets and 1.2 for outer ones.     

Asymptotic series for planetary motion in periodic terms in three dimensions

For the planetary case of the gravitationaln-body problem in three dimensions, a sequence of Lie series contact transformations is used to construct asymptotic series representations for the canonical parameters of the instantaneous orbits in a Jacobi formulation. The series contain only periodic terms, the frequencies being linear combinations of those of the planetary orbits and those of the secular variations of the apses and nodes, and the series are in powers of the masses of the planets in terms of that of the primary, and of a quantity of the order of the excursions of the eccentricities and inclinations of the orbits. The treatment avoids singularities for circular and coplanar orbits. It follows that the major axes are given by series of periodic terms only, to all orders in the planetary masses.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.     

A magnetic dynamo in the moon?

The existence of fossil lunar magnetism has caused speculation that the Moon had, at one time, an internally produced dynamo magnetic field. Quantitative analysis of this idea, constrained by the largest iron lunar core compatible with observations, implies that the Moon would have had to rotate faster than its breakup angular velocity in order to support a dynamo magnetic field.A paper presented at the Lunar Science Institute Conference on Geophysical and Geochemical Exploration of the Moon and Planets, January 10–12, 1973.     

Thermoluminescence of the lunar surface

Observations of the lunar luminescence are reported for a dozen of specific Moon features using the line-depth method with a high resolution spectroscopic technique. The data indicate a variation of the Moon proper emission as a function of the phase angle which is interpreted as a proof of the thermoluminescent origin of this emission.     

New orbital elements for Moon and planets

The results of a simultaneous solution for the orbital elements of Moon and planets are given and their derivation is discussed. A modern Cowell integrator is used for orbit computations, and least-squares fits are made to some 40000 optical observations taken since 1913. The model includes relativistic terms, the leading zonal harmonics of Earth and Moon, the precession of the lunar equator, and the tidal couple between Earth and Moon. The tidal term in the Moon's mean longitude is found to be –19±4 per century squared. The solution also yields an extrapolation of the atomic time scale back to 1912.5. At that time, the difference between atomic and ephemeris time is about 6±2 s. Lunar declinations observed by the Washington transit circles, after receiving limb corrections and thus with respect to the center of Watts' reference sphere, are smaller than computed values by 0.33±0.01. It is found that solar oblateness cannot quite be determined with optical data covering about 50 yr, butJ ₂ is unlikely to be much larger than 10^–5. The advance of Mercury's perihelion is verified to within our resolution of 2 per century to match that predicted by Einstein.The solution presented here is believed to be the only simultaneous improvement of the orbits of Moon and planets. This simultaneity is found to be an essential feature in separating the Moon's mean motion, the lunar tidal deceleration, and the corrections to the Earth rotation rate. It is now possible to refer all astronomical events of the past 60 yr to a time with uniform rate, namely the atomic clock system. Considering the long baseline, this model should facilitate the prediction of fast variables, such as the lunar longitude, with considerably increased confidence. The planetary orbital elements compete with efforts of similar scope and accuracy at the Massachusetts Institute of Technology and the Jet Propulsion Laboratory.     

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.