Hypothesis of variable g and the secular accelerations of the Sun and the Moon

It is known that the observed secular accelerations of the Sun and Moon are not consistent with the tidal interactions of the Earth with the Sun and Moon. Following Dicke, the hypothesis of variable constant of gravity is adopted and expressions for the accelerations are derived. It is shown that if the theoretical ratio of the acceleration is equated the observed one, a unique value for —/G can be calculated. Adopting the accelerations obtained by Fotheringham, Newton, Muller and Stephenson, and Stephenson, it is found that — /G ranges from 1.4 × 10^–11 to 3.3 × 10^–11 yr^–1. This estimate is consistent with the one based upon the comparison of the lunar accelerations measured with respect to atomic and ephemis times.     

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.     

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.     

On the accelerations of the Moon and Sun,the constant of gravitation,and the origin of mountains

In a previous paper Lyttleton (1976) has shown that the apparent secular accelerations of the Sun and Moon, as given by de Sitter, can be largely explained if the Earth is contracting at the rate required by the phase-change hypothesis for the nature of the core. More reliable values for these accelerations have since become available which warrant a redetermination of the various effects concerned on the basis of constantG, and this is first carried out in the present paper. The lunar tidal couple, which is the same whetherG is changing or not, is found to be (4.74±0.38)×10²³ cgs, about three-quarters that yielded by the de Sitter values, while within the theory the Moon would take correspondingly longer to reach close proximity to the Earth at about 1.5×10⁹ years ago.The more accurate values of the accelerations enable examination to be made of the effects that a decreasingG would have, and it is shown that a valueG/G=–3×10^–11 yr^–1 can be weakly satisfied compared with the close agreement found on the basis of constantG, while a value as large numerically asG/G=–6×10^–11 yr^–1 seems to be definitely ruled out. On the iron-core model, an intrinsic positive component of acceleration of the angular velocity cannot be reconciled at all with the secular accelerations even for constantG, and far less so ifG is decreasing at a rate suggested by any recent cosmological theory.ItG=0, the amount of contraction available for mountain-building would correspond to a reduction of surface area of about 49×10⁶ km² and a volume to be redistributed of 160×10⁹ km³ if the time of collapse were 2.5×10⁹ years ago. For earlier times, the values are only slightly reduced. IfG/G=–3×10^–11 yr^–1, the corresponding values are 44×10⁶ km² and 138×10⁹ km³ for collapse at –2.5×10⁹ yr, and not importantly smaller at 38×10⁶ km² and 122×10⁹ km³ for collapse at –4.5×10⁹ yr. Any of these values would suffice to account in order of magnitude for all the eras of mountain-building. An intense brief period of mountain-building on an immense scale would result from the Ramsey-collapse at whatever time past it may have occurred.     

Relation of a contracting earth to the apparent accelerations of the sun and moon

The tidal theory of the evolution of the lunar orbit has remained inconsistent with the observational values of the apparent secular accelerations of the Sun and Moon since it was first developed by Jeffreys in 1920. Allowance for a changing moment of inertia of the Earth enables the discrepancy to be completely removed if a decrease is occurring at a rate of just about the amount already required by the phase-change theory of the nature of the terrestrial core. The agreement of the resulting theory with the latest determinations of the lunar acceleration increases confidence in the phase-change hypothesis. On the other hand the theory renders it most unlikely that a changing constant of gravitation will prove necessary to account for the observations. On the present theory of itself the Moon would have been extremely close to the Earth only about 10⁹ yr ago which suggests that some additional process may at times have influenced the lunar orbit.     

Heat conduction within an elastic Earth

We use theoretical results derived in a previous paper (Lanzano, 1986) to numerically evaluate the temperature profile and radial deformation within a spherical, elastic Earth due to heat generated by the decay of radiogenic elements.We consider only the Uranium family and have assumed the diffusivity of the silicate mantle to be K = 8 × 10^–3 cm² s^–1, the Poisson elastic ratio to be = 0.25 and the coefficient of thermal expansion to be = 2 × 10^–5 (deg)^–1. Our series solutions when applied to the interiors of the Moon, Mercury, and Mars yield results in agreement with Kopal's (1963) evaluations.     

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     

The Earth-figure perturbations in the Lunar theory

We compute the perturbations on the motion of the Moon due to the shape of the Earth. The zonal terms inJ ₂,J ₃, andJ ₄ are considered. The accuracy is estimated at 3×10^–5 and the results compared with previous theories.     

A note on tidal deceleration of the moon

The discrepancy between the observed apparent acceleration of the Moon in longitude (1) and the actual lunar laser ranging data (3), (4) is of the order of ~ 9 × 10^–23 rad s^–2. It cannot be explained by the rms errors in (1) and (3), (4); processes connected with the internal Earth's dynamics and accelerating the Earth in its rotation might be responsible for the phenomenon, leading to the decreasing of the principal moment of the Earth's inertia ~ – 3.2 × 10²⁹m² kg cy^–1.     

The Hard X-ray Telescope (HXT) for the SOLAR-A mission

The Hard X-ray Telescope (HXT) is a Fourier-synthesis imager; a set of spatially-modulated photon count data are taken from 64 independent subcollimators and are Fourier-transformed into an image by using procedures such as the maximum entropy method (MEM) or CLEAN. The HXT takes images of solar flares simultaneously in four energy bands, nominally 15 (or 19)–24, 24–35, 35–57, and 57–100 keV, with an ultimate angular resolution as fine as 5 arc sec and a time resolution 0.5 s. Each subcollimator has a field of view wider than the solar disk. The total effective area of the collimator/detector system reaches 70 cm², about one order of magnitude larger than that of the HINOTORI hard X-ray imager. Thanks to these improvements, HXT will for the first time enable us to take images of flares at photon energies above 30 keV. These higher-energy images will be compared with lower-energy ones, giving clues to the understanding of nonthermal processes in solar flares, i.e., the acceleration and confinement of energetic electrons. It is of particular importance to specify the acceleration site with regard to the magnetic field figuration in a flaring region, which will be achieved by collaborative observations between HXT and the Soft X-ray Telescope on board the same mission.After the launch the name of SOLAR-A has been changed to YOHKOH.     

The velocity structure of the lunar crust

Seismic refraction data, obtained at the Apollo 14 and 16 sites, when combined with other lunar seismic data, allow a compressional wave velocity profile of the lunar near-surface and crust to be derived. The regolith, although variable in thickness over the lunar surface, possesses surprisingly similar seismic properties. Underlying the regolith at both the Apollo 14 Fra Mauro site and the Apollo 16 Descartes site is low-velocity brecciated material or impact derived debris. Key features of the lunar seismic velocity profile are: (i) velocity increases from 100–300 m s^–1 in the upper 100 m to 4 km s^–1 at 5 km depth, (ii) a more gradual increase from 4 km s^–1 to 6 km s^–1 at 25 km depth, (iii) a discontinuity at a depth of 25 km and (iv) a constant value of 7 km s^–1 at depths from 25 km to about 60 km. The exact details of the velocity variation in the upper 5 to 10 km of the Moon cannot yet be resolved but self-compression of rock powders cannot duplicate the observed magnitude of the velocity change and the steep velocity-depth gradient. Other textural or compositional changes must be important in the upper 5 km of the Moon. The only serious candidates for the lower lunar crust are anorthositic or gabbroic rocks.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973.     

Period variation and the new light curves of V471 Tauri

The light curve and period variation of the eclipsing system V471 Tau is discussed. The migration period of 191 days obtained recently by Ibanolu agrees well with the new observations. The period decreases by about one second per century, which may correspond to a mass transfer of 1×10^–7 solar mass. Furthermore, the O–C diagram shows a noticeable sine curve super-imposed on the parabolic variation. The period of the sine curve is about 3.1 years.     

The phase of particle acceleration in the flare development

Evidence is given that the particle acceleration in flares is confined to the initial phase of the flare development preceding the H flare maximum and lasting for less than 10 min. The impulsive acceleration process is confined to a relatively small limited volume of about 5 × 10²⁷ cm³ in the region of highest magnetic gradient in the flare, and its size represents about 0.05 or less of the total extent of the hot condensation which produces the soft X-ray and gradual microwave bursts. About one in fifty particles in this volume is accelerated to energy exceeding 100 keV, the total particle density being 10¹⁰ cm^–3. The accelerated electrons produce the impulsive hard X-ray burst, but synchrotron losses greatly reduce the number of relativistic electrons participating in the bremsstrahlung process. Protons above 20 MeV penetrate to the lowest chromosphere and upper photosphere and temporarily increase the temperature in the bombarded region. As the result a flash of continuous emission appears, which should be most expressive below 1527 Å. The associated white-light emission shows the bottom of the region where the impulsive acceleration process occurs.     

Direct and indirect capture of near-Earth asteroids in the Earth–Moon system

Near-Earth asteroids have attracted attention for both scientific and commercial mission applications. Due to the fact that the Earth–Moon \(\hbox {L}_{1}\) and \(\hbox {L}_{2}\) points are candidates for gateway stations for lunar exploration, and an ideal location for space science, capturing asteroids and inserting them into periodic orbits around these points is of significant interest for the future. In this paper, we define a new type of lunar asteroid capture, termed direct capture. In this capture strategy, the candidate asteroid leaves its heliocentric orbit after an initial impulse, with its dynamics modeled using the Sun–Earth–Moon restricted four-body problem until its insertion, with a second impulse, onto the \(\hbox {L}_{2}\) stable manifold in the Earth–Moon circular restricted three-body problem. A Lambert arc in the Sun-asteroid two-body problem is used as an initial guess and a differential corrector used to generate the transfer trajectory from the asteroid’s initial obit to the stable manifold associated with Earth–Moon \(\hbox {L}_{2}\) point. Results show that the direct asteroid capture strategy needs a shorter flight time compared to an indirect asteroid capture, which couples capture in the Sun–Earth circular restricted three-body problem and subsequent transfer to the Earth–Moon circular restricted three-body problem. Finally, the direct and indirect asteroid capture strategies are also applied to consider capture of asteroids at the triangular libration points in the Earth–Moon system.     

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.     

Tungsten isotopes in ferroan anorthosites: Implications for the age of the Moon and lifetime of its magma ocean

New W isotope data for ferroan anorthosites 60025 and 62255 and low-Ti mare basalt 15555 show that these samples, contrary to previous reports [Lee, D.C., et al., 1997. Science 278, 1098-1103; Lee, D.C., et al., 2002. Earth Planet. Sci. Lett. 198, 267-274], have a W isotope composition that is indistinguishable from KREEP and other mare basalts. This requires crust extraction on the Moon later than ∼60 Myr after CAI formation, consistent with ¹⁴⁷Sm-¹⁴³Nd ages for ferroan anorthosites. The absence of ¹⁸²Hf-induced ¹⁸²W variations in the Moon is consistent with formation of the Moon at after CAI formation that has been inferred based on the indistinguishable ¹⁸²W/¹⁸⁴W ratios of the bulk Moon and the bulk silicate Earth. The uncertainties on the age of the Moon and the age of the oldest lunar samples currently hamper a precise determination of the duration of magma ocean solidification and are consistent with both an almost immediate crystallization and a more protracted timescale of ∼100 Myr.     

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.     

Upper limit on the steady emission of the 2.223 MeV neutron capture γ-ray line from the Sun

Data accumulated by the Solar Maximum Mission Gamma Ray Spectrometer (GRS) have been searched for evidence of the 2.223 MeV neutron capture line from the Sun, outside the times of -ray-emitting solar flares. Background-corrected spectra accumulated over 3-day intervals between 1980 and 1989 show no evidence of the line. Upper limits are reported separately for periods of high and low solar activity.A conservative 3 upper limit of 5.7 × 10^–5 (cm² s)^–1 is placed on the steady flux in the 2.223 MeV line during inactive periods, which is nearly two orders of magnitude lower than previously published results. After correction for limb darkening of the line emission from off-center positions, this upper limit becomes 7.1 × 10^–5 (cm²s)^–1. Our 3 upper limit on the steady flux in the line during periods of high solar activity is 6.9 × 10^–5 (cm² s)^–1, or 8.6 × 10^–5 (cm² s)^–1 after correction for limb darkening. Our results imply that the quiescent solar corona cannot be heated by ions accelerated above 1 MeV in microflares (or a continuous acceleration process), so long as the ion energy spectrum is similar to that measured in large flares. We also use our results to derive the rate of tritium production at the solar surface; our upper limit of 9 nuclei (cm² s)^–1 is about a factor of 9 below the upper limit from searches for ³H in the solar wind. We place upper limits of the order 10³³ on the number of energetic (> 30 MeV) protons which can be stored in active regions prior to being released in solar flares, which imply that the strongest observed flares cannot be produced by such a mechanism.     

On the use of analytical theories in the calculation of precession-nutation

We present the use of the analytical solutions of the planets and of the Moon's motion in the determination of the quantities which relate the barycentric and the geocentric coordinate systems and of the expressions of precession-nutation. The computation of the precession and nutation quantities are built with the analytical theories of the motion of the Moon, the Sun and the planets of the Bureau des longitudes. We take into account the influence of the Moon, the Sun and all the planets on the potential of the Earth limited to C _j,0 for j from 2 to 5, C _2,2, S _2,2, C _3, S _3, , for from 1 to 3 and C _4,1, S _4,1. We determine the 3 Euler angles , , and 2 calculating the components of the torque of the external forces with respect to the geocenrer in the case of the rigid Earth. The equations are solved by iterations and so are taken into account the nutations-on-the-nutations effects. We have determined the analytical variations of the angles and w fixing the equator with respect to the ecliptic J2000. We find, in w, a secular term of –26.5026 mas per century. The analytical solution of the precession-nutation has been compared to a numerical integration over the time span 1900–2050. The differences do not exceed 16 µas for and 12 µas for .     

Bombardment as a cause of the lunar asymmetry

A comparison of the lunar frontside gravity field with topography indicates that low-density ( 2.9 g cm^–3) types of rock form a surface layer or crust of variable thickness: 40-60 km beneath terrae; 20-40 km beneath non-mascon maria; 0-20 km beneath mascon maria. The observed offset between lunar centers of mass and figure is consistent with farside crustal thicknesses of 40-50 km, similar to frontside terra thicknesses.The Moon is asymmetric in crustal thickness, and also in the distribution of maria and gamma radioactivity. Early bombardment of the Moon by planetesimals, in both heliocentric and geocentric orbits, is examined as a possible cause of the asymmetries. The presence of a massive companion (Earth) causes a spin-orbit coupled Moon to be bombarded non-uniformly. The most pronounced local concentration of impacts would have occurred on the west limb of the Moon, when it orbited close to the Earth, if low-eccentricity heliocentric planetesimals were still abundant in the solar system at that time.A very intense bombardment of this type could have redistributed crustal material on the Moon, thinning the west limb crust appreciably. This would have caused a change in position of the principal axes of inertia, and a reorientation of the spin-orbit coupled Moon such that the thinnest portion of its crust turned toward one of the poles. Erupting lavas would have preferentially flooded such a thin-crusted, low-lying area. This would have caused another readjustment of principal moments, and a reorientation of the Moon such that the mare areas tipped toward the equator. The north-south and nearside-farside asymmetries of mare distribution on the present Moon can be understood in terms of such a history.Paper dedicated to Prof. Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.