The evolution of the Earth-Moon system

The tidally-induced couple acting on the Moon, due to friction between the oceans and their beds, is calculated as a function of the Earth-Moon separation. The function is found to be proportional to 1+d/R ³ , and not the previously used 1/R ⁶. By use of this new function it is found that the present rate of lunar recession gives an acceptable history for the system if it is assumed the Moon was initially in a close geo-stationary orbit 4 billion years ago, when perturbed by the condensation of the Earth's core.     

Tidal-friction theory of the Earth-Moon system

The standard discussion of tidal friction in the Earth-Moon system has been that given by Jeffreys in successive editions ofThe Earth over the past several decades. It is herein shown to contain several erros vitiating its results. The dynamical equation utilised for finding the rate of change of angular velocity of the Earth fails to take account of the fact that the moment of inertia of the Earth may be changing with time, and all subsequent equations which depend on this are incorrect as a result. Simple equations have been left unsolved that ought to have been solved, and the alleged numerical conclusions in no way follow from the values set down initially for the observed apparent secular accelerations of the Moon and Sun.The revised dynamical equations are shown to enable the lunar and solar tidal couples to conform to theory, and may imply that the moment of inertia of the Earth is decreasing at a non-negligible rate. Recognition of this is the key to the whole problem. The only available hypothesis providing adequate contraction is that following from the phase-change theory of the nature of the terrestrial core, and the value of the rate of decrease of moment of inertia calculated from this is in close agreement with that implied by modern improved values of the secular accelerations.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Note on tidal evolution of the Earth-Moon system

The need is pointed out of a re-discussion of the past tidal evolution of the Earth-Moon system as a boundary-value problem on the time-scale indicated by radiometric dating of lunar soils returned by successive space missions from different localities on the Moon's surface.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Astronomical evidence concerning non-gravitational forces in the Earth-Moon system

The most conspicuous effects of non-gravitational forces in the Earth-Moon system are the accelerations of the Earth's spin and of the Moon's mean angular velocity. Evidence indicates that the present acceleration of the Moon is between –20 and –52 s of arc per century per century and that the present average acceleration of the Earth is between –5 and –23 parts in 10⁹ per century. Over the past 2000 yr, the average for the Moon has been about –42 s per century per century and for the Earth has been about –28 parts in 10⁹ per century; these values are probably correct within 10%. Evidence that does not involve any assumptions about the present values shows strongly that there was a square wave in the accelerations that lasted from about 700–1300, and that the accelerations were different by a factor of perhaps 5 during the time of this wave from what they were at neighboring times.An effect that seems to be changing the obliquity of the ecliptic has been reported in recent literature, on the basis of data obtained within the past century. The effect amounts to about 1/4 s of arc per century if it is real. Older data are not accurate enough to give information about an effect this small.There are no satisfactory explanations of the accelerations. Existing theories of tidal friction are quite inadequate.Paper presented at the AAAS Symposium on the Early History of the Earth and Moon in Philadelphia on 28 December 1971.     

Lunar mare domes: Classification and modes of origin

A classification of over 200 lunar mare domes shows that they have two major modes of occurrence: (1) low, flat, generally circular structures with convex shapes, slopes less than about 5°, and displaying summit craters, and (2) irregular structures often adjacent to highland regions and rarely containing summit craters. On the basis of morphologic and morphometric similarities, the first mode of occurrence appears to be analogous to small terrestrial shield volcanoes, and to represent primary volcanic constructs, while the second class of domes appears to result from secondary volcanic effects (flooding of highland material to produce kipukas and draping of lavas to produce irregular dome-like topography).Domes comparable to small shield volcanoes generally range from 3–17 km in diameter and up to several hundred meters in height and occur predominantly in groupings in the lunar equatorial region in northeast Tranquillitatis (Cauchy area), between Kepler and Copernicus (Hortensius area), and in the Marius Hills. In the Marius Hills, domes generally lack summit craters and have a rough surface texture formed in part by superposed cones and steep-sided flows. Elsewhere, domes representing volcanic sources are smooth-surfaced and usually contain a summit crater. These features are similar in general morphology to small terrestrial lava shields. They are generally intermediate in volume, slope, and height between small shields of terrestrial basaltic plains (such as the Snake River Plains) and larger Icelandic shields. Summit craters on lunar domes are considerably larger than craters on terrestrial shields of comparable diameters, apparently due to a combination of factors, including vent enlargement during extrusion, possibly higher lunar extrusion rates, different amounts of collapse, and impact erosion.Most vent-related domes appear to be associated with, and are thus approximately the same age as, surrounding lava plains, although relationships in specific areas have not yet been established. On the basis of age ranges of mare deposits established by Apollo samples, mare vent-related domes formed over an approximately one billion year period starting about 3.7 b.y. ago. Extrusion rates were apparently relatively low compared to the very high values characteristic of flows associated with major lunar sinuous rilles and terrestrial flood basalts, but may have been relatively high compared to similar terrestrial shields. Large shield volcanoes equivalent to the terrestrial Hawaiian-type or to the martian edifices such as Olympus Mons, do not occur on the Moon. Lack of these features may be due to the low viscosities and high effusion rates typical of many lunar eruptions and the lack of continuous eruptions from single sources.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

On the dichotomy in the Earth-Moon system restricted three-body problem

We apply the numerical technique of Poincare surface of section to investigate the dichotomy present in the Earth-Moon system, considering the framework of planar, circular, restricted three-body problem. A study on the transition of quasi-periodic orbits (oscillatory type dichotomy) present at the Jacobi constant C=2.85 shows that the dichotomy discussed here exist not at a particular value of the mass ratio and the Jacobi constant. It is observed that as C increases, the range of mass ratio at which the dichotomy pertains increases, even though the mass ratio at which the transition of orbits takes place decreases.     

Current bombardment of the Earth-Moon system: Emphasis on cratering asymmetries

We calculate the current spatial distribution of projectile delivery to the Earth and Moon using numerical orbital dynamics simulations of candidate impactors drawn from a debiased Near-Earth Object (NEO) model. We examine the latitude distribution of impactor sites and find that for both the Earth and Moon there is a small deficiency of time-averaged impact rates at the poles. The ratio between deliveries within 30° of the pole to that of a 30° band centered on the equator is small for Earth (<5%) (0.958±0.001) and somewhat greater for the Moon (∼10%) (0.903±0.005). The terrestrial arrival results are examined to determine the degree of AM/PM asymmetry to compare with the PM excess shown in meteorite fall times. We find that the average lunar impact velocity is 20 km/s, which has ramifications in converting observed crater densities to impactor size distributions. We determine that current crater production on the leading hemisphere of the Moon is 1.28±0.01 that of the trailing when considering the ratio of craters within 30° of the apex to those within 30° of the antapex and that there is virtually no nearside-farside asymmetry, in agreement with observations of rayed craters. As expected, the degree of leading-trailing asymmetry increases when the Moon's orbital distance is decreased.     

Long term evolution of distant retrograde orbits in the Earth-Moon system

This work studies the evolution of several Distant Retrograde Orbits (DROs) of varying size in the Earth-Moon system over durations up to tens of millennia. This analysis is relevant for missions requiring a completely hands off, long duration quarantine orbit, such as a Mars Sample Return mission or the Asteroid Redirect Mission. Four DROs are selected from four stable size regions and are propagated for up to 30,000 years with an integrator that uses extended precision arithmetic techniques and a high fidelity dynamical model. The evolution of the orbit’s size, shape, orientation, period, out-of-plane amplitude, and Jacobi constant are tracked. It has been found that small DROs, with minor axis amplitudes of approximately 45,000 km or less decay in size and period largely due to the Moon’s solid tides. Larger DROs (62,000 km and up) are more influenced by the gravity of bodies external to the Earth-Moon system, and remain bound to the Moon for significantly less time.     

Near-resonance tidal evolution of the Earth-Moon system influenced by orbital-scale climate change

We build a conceptual coupled model of the climate and tidal evolution of the Earth-Moon system to find the influence of the former on the latter. An energy balance model is applied to calculate steady-state temperature field from the mean annual insolation as a function of varying astronomical parameters. A harmonic oscillator model is applied to integrate the lunar orbit and Earth's rotation with the tidal torque dependent on the dominant natural frequency of ocean. An ocean geometry acts as a bridge between temperature and oceanic frequency. On assumptions of a fixed hemispherical continent and an equatorial circular lunar orbit, considering only the 41 kyr periodicity of Earth's obliquity ε and the M2 tide, simulations are performed near tidal resonance for 106 yr. It is verified that the climate can influence the tidal evolution via ocean. Compared with the tidal evolution with constant ε, that with varying ε is slowed down; the EarthMoon distance oscillates in phase with ε before the resonance maximum but exactly out of phase after that;the displacement of the oscillation is in positive correlation with the difference between oceanic frequency and tidal frequency.     

Counterglow from the Earth-Moon libration points

Results from the OSO-6 Rutgers Zodiacal Light Analyzer experiment show photometric perturbations above the background in the anti-Sun line of sight. Sixteen successive lunations were examined, and the accumulated perturbations show a maximum value in the direction of the L₄ and L₅ Earth-Moon libration points. This is interpreted as a counterglow from a cloud of particles at the libration points. The average brightness of these libration clouds is 20 S₁₀ Vis. The average angular size of the libration clouds is approximately 6 degrees. Their position varies from one lunation to the next, within an ellipsoidal zone centered on the libration point direction, with its semi-major axis, of approximately 6 degrees, nominally in the ecliptic and its semi-minor axis, of approximately 2 degrees perpendicular to the ecliptic. The position of these clouds with respect to the Lagrangian L₄ and L₅ points, is towards the Moon in the northern summer and away from the Moon in the northern winter.     

Temporal fluctuations and anisotropy of the micrometeoroid flux in the Earth-Moon system measured by HEOS 2

The HEOS detector measures the mass and speed of micrometeoroids in the Earth-Moon system. They are detected by the plasma produced by particle impacts on the sensor. During 2 yr of data collection 384 particles have been registered. As shown earlier (COSPAR 1973), they can be divided into 3 categories according to their temporal distribution: particles that are (1) randomly distributed or (2) appear in “groups” or (3) appear in “swarms” In this paper the origin of the groups and swarms is discussed. For this purpose the article orbits with respect to the Earth and the Moon were traced back. The results imply a lunar origin of the groups, whereas the swarms are correlated with the vicinity of the Earth. In addition, the dependence of the cumulative flux upon the detector's viewing direction indicates clearly an anisotropic particle flux.     

Northwest Africa 773: Lunar origin and iron‐enrichment trend

Abstract— The meteorite Northwest Africa 773 (NWA 773) is a lunar sample with implications for the evolution of mafic magmas on the moon. A combination of key parameters including whole‐rock oxygen isotopic composition, Fe/Mn ratios in mafic silicates, noble gas concentrations, a KREEP‐like rare earth element pattern, and the presence of regolith agglutinate fragments indicate a lunar origin for NWA 773. Partial maskelynitization of feldspar and occasional twinning of pyroxene are attributed to shock deformation. Terrestrial weathering has caused fracturing and precipitation of Carich carbonates and sulfates in the fractures, but lunar minerals appear fresh and unoxidized. The meteorite is composed of two distinct lithologies: a two‐pyroxene olivine gabbro with cumulate texture, and a polymict, fragmental regolith breccia. The olivine gabbro is dominated by cumulate olivine with pigeonite, augite, and interstitial plagioclase feldspar. The breccia consists of several types of clasts but is dominated by clasts from the gabbro and more FeO‐rich derivatives. Variations in clast mineral assemblage and pyroxene Mg/(Mg + Fe) and Ti/(Ti + Cr) record an igneous Fe‐enrichment trend that culminated in crystallization of fayalite + silica + hedenbergite‐bearing symplectites. The Fe‐enrichment trend and cumulate textures observed in NWA 773 are similar to features of terrestrial ponded lava flows and shallow‐level mafic intrusives, indicating that NWA 773 may be from a layered mafic intrusion or a thick, differentiated lava flow. NWA 773 and several other mafic lunar meteorites have LREE‐enriched patters distinct from Apollo and Luna mare basalts, which tend to be LREE‐depleted. This is somewhat surprising in light of remote sensing data that indicates that the Apollo and Luna missions sampled a portion of the moon that was enriched in incompatible heatproducing elements.     

Photographic observations of the cloud in the neighbourhood of libration point L 5 of the Earth-Moon system

The paper contains results of three-colour photographic observations of positions and brightness of the cloud in the vicinity of the Earth-Moon libration point L₅. The real character of the images obtained is confirmed by an agreement of their positions on different plates exposed at the same time. The colours of the cloud obtained are essentially different from those of the counterglow. The clouds appeared to be much redder than the counterglow, which may indicate that the particles constituting them are of different nature than those causing the counterglow.     

Lunar Beagle and Lunar Astrobiology

The study of the elements and molecules of astrobiological interest on the Moon can be made with the Gas Analysis Package (GAP) and associated instruments developed for the Beagle 2 Mars Express Payload. The permanently shadowed polar regions of the Moon may offer a unique location for the “cold-trapping” of the light elements (i.e. H, C, N, O, etc.) and their simple compounds. Studies of the returned lunar samples have shown that lunar materials have undergone irradiation with the solar wind and adsorb volatiles from possible cometary and micrometeoroid impacts. The Beagle 2’s analytical instrument package including the sample processing facility and the GAP mass spectrometer can provide vital isotopic information that can distinguish whether the lunar volatiles are indigenous to the moon, solar wind derived, cometary in origin or from meteoroids impacting on the Moon. As future Lunar Landers are being considered, the suite of instruments developed for the Mars Beagle 2 lander can be consider as the baseline for any lunar volatile or resource instrument package.     

Solar and planetary destabilization of the Earth-Moon triangular Lagrangian points

In the restricted circular three-body problem, two massive bodies travel on circular orbits about their mutual center of mass and gravitationally perturb the motion of a massless particle. The triangular Lagrange points, L₄ and L₅, form equilateral triangles with the two massive bodies and lie in their orbital plane. Provided the primary is at least 27 times as massive as the secondary, orbits near L₄ and L₅ can remain close to these locations indefinitely. More than 2200 cataloged asteroids librate about the L₄ and L₅ points of the Sun-Jupiter system, and five bodies have been discovered around the L₄ point of the Sun-Neptune system. Small satellites have also been found librating about the L₄ and L₅ points of two of Saturn's moons. However, no objects have been discovered around the Earth-Moon L₄ and L₅ points. Using numerical integrations, we show that orbits near the Earth-Moon L₄ and L₅ points can survive for over a billion years even when solar perturbations are included, but the further addition of the far smaller perturbations from other planets destabilize these orbits within several million years. Thus, the lack of observed objects in these regions cannot be used as a constraint on Solar System formation, nor on the tidal evolution of the Moon's orbit.     

今后几年的月球探测和月球科学

近年来月球探测已经进入了一个全新的时代。特别是 1 990年以来 ,多个月球探测计划已经被成功实现 ,而且另外还有多个探测计划也在准备当中 ,并将在未来的几年内发射升空。在这种背景之下 ,中国的航天机构和有关的科学家也开始积极酝酿和开发自己的月球探测计划。这些月球探测计划将利用卫星上搭载的各种仪器探测和测量月球的地质和地理特性、化学成分和矿物组成、月球物理学特征以及包含地球大气在内的地月空间环境和行星际空间环境 ;进一步研究月球的起源和演化 ,探明月面环境 ,研究太阳等离子体物理 ,提供月面天文台和月面长期科研基地的候选地址 ,调查月球上的可利用资源 ,为将来开发月球提供充实的背景资料。参与新一轮的月球探测同样也为中国天文学研究带来了新的机会。     

潮汐能量耗散与地月系统演化

对地月系统而言, 在很大程度上角动量守恒是正确的. 地月距离的变化主要是受到月球引起的潮汐能量耗散的影响. 根据月球的平均运动和它的长期加速度, 就可以计算出月潮能量耗散的数值. 海洋是潮汐能量耗散的主要区域. 由于潮汐的高度正比于月球对潮汐隆起的万有引力, 由此可导出总的月球潮汐摩擦力正比于月球平均运动的平方. 如果采用月球平均加速度数值-20.72$''\cdot$cy^-2, 就可以推算出35亿年来地月之间的距离以及回归年日数和朔望月日数的演化. 此理论结果与古生物钟的数据进行比对, 两者符合较好.     

Co-accretion of the Earth-Moon system after the giant impact: reduction of the total angular momentum by lunar impact ejecta

Though the Moon is considered to have been formed by the so-called giant impact, the mass of the Earth immediately after the impact is still controversial. If the Moon was formed during the Earth's accretion, a subsequent accretion of residual heliocentric planetesimals onto the protoearth and the protomoon must have occurred. In this co-accretion stage, a significant amount of lunar-impact-ejecta would be ejected to circumterrestrial orbits, since the mean impact velocity of the planetesimals with the protomoon is much larger than the escape velocity of the protomoon. Orbital calculations of test particles ejected from the protomoon, whose semimajor axis is smaller than that of the present Moon, reveal that most of the particles escaping from the protomoon also escape from the Hill sphere of the protoearth and reduce the planetocentric angular momentum of the primordial Earth-Moon system. Using the results of the ejecta simulations, we investigate the evolution of the mass ratio and the total angular momentum (Earth's spin angular momentum + Moon's orbital angular momentum) of the Earth-Moon system during the co-accretion. We find that the mass of the protomoon is almost constant or rather decreases and the total angular momentum decreases significantly, if the random velocity of planetesimals is as large as the escape velocity of the protoearth. On the other hand, if the random velocity is the half of the escape velocity of the protoearth, the mass ratio is kept to be almost as large as the present value and the decrease of the total angular momentum is not so significant. Comparing with the results of giant impact simulations, we find that the mass of the protoearth immediately after the Moon-forming impact was 0.7-0.8 times the present value if the impactor-to-target mass ratio was 3:7, whereas the giant impact occurred almost in the end of the Earth's accretion if the impactor-to-target mass ratio was 1:9.