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.     

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.     

Comments on the figure of the moon from apollo landmark tracking

Optical observations were made from the orbiting spacecraft to craters on the lunar surface during Apollo missions 8, 10, 11, 12, 14, and 15. Very accurate selenographic locations for 31 craters have been obtained from these data. The estimated radius values, with respect to the center of mass of the Moon, for the near side maria were smaller than the nominally accepted value of 1738 km. Gross figure of the Moon estimates were obtained for both a sphere and a constrained ellipsoid. These data appear to provide some proof that there is a displacement between the center of figure and the center of mass of the Moon.     

On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon

From the observations of the gravitational field and the figure of the Moon, it is known that its center of mass (briefly COM) does not coincide with the center of figure (COF), and the line “COF/COM” is not directed to the center of the Earth, but deviates from it to the South–East. Here we study the deviation of the lunar COM to the East from the mean direction to Earth.At first, we consider the optical libration of a satellite with synchronous rotation around the planet for an observer at a point on second (empty) orbit focus. It is found that the main axis of inertia of the satellite has asymmetric nonlinear oscillations with amplitude proportional to the square of the orbit eccentricity. Given this effect, a mechanism of tidal secular evolution of the Moon’s orbit is offered that explains up to \(20\%\) of the known displacement of the lunar COM to the East. It is concluded that from the alternative—evolution of the Moon’s orbit with a decrease or increase in eccentricity—only the scenario of evolution with a monotonous increase in orbit eccentricity agrees with the displacement of lunar COM to the East. The precise calculations available confirm that now the eccentricity of the lunar orbit is actually increasing and therefore in the past it was less than its modern value, \(e = 0.0549\).To fully explain the displacement of the Moon’s COM to the East was deduced a second mechanism, which is based on the reliable effect of tidal changes in the shape of the Moon. For this purpose the differential equation which governs the process of displacement of the Moon’s COM to the East with inevitable rounding off its form in the tidal increase process of the distance between the Earth and the Moon is derived. The second mechanism not only explains the Moon’s COM displacement to the East, but it also predicts that the elongation of the lunar figure in the early epoch was significant and could reach the value \(\varepsilon\approx0.31\). Applying the theory of tidal equilibrium figures, we can estimate how close to the Earth the Moon could have formed.     

An Estimate on the Lunar Figure

In 1799 Laplace discovered that the three principal moments of the Moon are not in equilibrium with the Moon's current orbital and rotational state. Some authors suggested that the Moon may carry a fossil figure. More than 3 billion years ago, the liquid Moon was closer to the Earth and revolved faster. Then the Moon migrated outwards and its rotation slowed down. During the early stage of this migration, the Moon was continually subjected to tidal and rotational stretching and formed into an ellipsoid. Subsequently the Moon cooled down and solidified quickly. Eventually, the solid Moon's lithosphere was stable and as a result we may see the very early lunar figure.     

Tectonic tests of proposed polar wander paths for Mars and the Moon

We have tested the polar wander paths recently proposed for Mars by Schultz and Lutz-Garihan and for the Moon by Runcorn through a comparison of the lithospheric stress field predicted for rapid global reorientations against observed tectonic features. We have employed the theory of Vening Meinesz and of Melosh to calculate the reorientation stresses, and we argue that the formation of normal faults or graben in broad regions surrounding the former rotation poles should be the minimum tectonic signature of a reorientation that generates lithospheric stresses in excess of the extensional strength of near-surface material. Such regions of normal faults are not present in the vicinity of the most recent proposed paleopoles for Mars, despite the large magnitude of the predicted shear stress (1–2 kbar). The minimum tectonic criterion would not be relaxed by invoking gradual polar wander or by considering the superposition of stresses associated with the global lithospheric response to the Tharsis rise. We conclude that polar wander of the magnitude and timing proposed by Schultz and Kutz-Garihan did not occur. It follows either that Tharsis has always been located near the Martian equator or that Tharsis began to dominate the nonhydrostatic figure prior to the end of heavy bombardment so that any tectonic signature of reorientation has since been obliterated by cratering. The predicted directions of stresses that would result from the most recent episode of proposed polar wander on the Moon, including stresses produced by reorientation of both the rotational and tidal figures, show little or no correspondence to observed tectonic features in the vicinity of the postulated nearside paleopole. The magnitude of the predicted reorientation stress is at most a few tens of bars, however, so that the tectonic test of polar wander on the Moon is inconclusive.     

Inhomogeneities in the internal structure of Mars and the Moon from data on their gravity fields

The interpretation of planetary anomalies in the gravity fields of Mars and the Moon in relationship to their inhomogeneous internal structure is considered. The Martian and lunar gravity field models up to order and degree 20, three-layer (crust, mantle, core) model parameters, and planetary parameters have been used as input data. Models of the three-dimensional density distribution have been constructed for Mars and the Moon. The maps of horizontal density inhomogeneities at depths of 50, 100, and 1700 km for Mars and 60, 100, and 1400 km for the Moon are interpreted.     

Young thrust-fault scarps in the highlands: evidence for an initially totally molten moon

Thermoelastic stress calculations show that if only the outer few hundred kilometers of the Moon was initially molten and if it had a cool interior, i.e., the magma ocean model of the Moon, the highlands should not have any young, compressional tectonic features. In contrast, if the Moon was initially totally molten, the highlands should have 10-km- scale, ?0.5- to 1 × 10⁹-year-old thrust faults. Observations using the Apollo panoramic imagery show that young thrust faults do exist in the highlands. Extrapolation of the data suggests that some 2000 thrust-fault scarps, whose average length is 9 km, are in the highlands. The fault scarps generally occur in series or complexes of four or five scarps. The average length of these complexes is 50 km; the largest observed complex is 120 km long. Extrapolation of the data suggests that there are about 400 such complexes. The ages of the scarps range from 60±30 to 680±250 my, with a possible bias of up to plus a factor of 2 or minus a factor of 4. These scarps are by far the youngest endogenic features on the Moon. The selenographical, size, age, morphological, and azimuth frequency distributions of the scarps can be explained by the effects of the kilobar-level thermoelastic stresses, the 100-bar-level tidal and rotational stresses, and influence by preexisting structures. These results show that the Moon has recently entered an epoch of late stage, global tectonism and favor the concept that the Moon was initially totally molten.     

A correlation between large-scale asymmetry of the Moon and the foci of moonquakes

Some evidence is produced to show that the Moon is asymmetric on the large scale as regards its figure, structure, composition, the distribution of tidal force, natural remnant magnetism and magnetic anomaly. This asymmetry produces the state of stress in the Moon, causing most foci of moonquakes to be located in the nearside of the Moon.     

Distortions of the Moon's figure due to the Earth

The second zonal and the second sectorial Stokes parameters of the Moon's gravitational field and/or the polar and equatorial flattenings of the lunar triaxial level ellipsoid have been explained by the tidal and rotational distortions due to the Earth. The Epoch at which the Moon's figure formation was finished has been estimated as 1.6 × 10⁹ y B. P. when the Earth-Moon distance was about 168 400 km and the orbital/rotational period of the Moon about 8 days.     

对月球形状的估算

1799年,Laplace发现月球的3个主惯量矩,与月球的轨道和自转状态并不相符.有些学者认为,这可能是现在的月球仍保留了早期的"化石"形状.大约在三十多亿年前,月球曾经离地球很近并且转得较快,然后月球逐渐迁移远离地球并且转动得慢了下来.在此迁移的较早时期,月球受到了引潮力和自转离心力的作用,成为一个椭球体.并且很快凝固.所幸的是,固态月球的岩石圈较为稳定,使我们现在仍然能够看到很早时期月球的形状.文中利用月球天平动参数以及引力场系数,计算了椭球体3个主向径a,b,c的长度和月球的平衡潮形状,得到如下3个结论:(1)开始时月球离地球是非常近的,大约在三十亿年前月球可能已经冷却和固化,现在的月球基本上保留了凝结时的形状.(2)证明了液态月球的潮汐形变是月球平衡潮高度的1.934倍.因此用月球引力场推算月球形状时,必需考虑到流体勒夫数hf=1.934的影响.(3)根据月球三个主轴a,6,c的长度之差,推算了月球临凝固时的月地距离为1.7455×1O8m,自转周期为3.652 day.从而推算出月球临凝固时的恒星月长度为8.34day.因此在月球凝结时,月球被锁定在与自转速率比为2:1的共振轨道上.     

Lander and scientific equipment for exploring of volatiles on the Moon

It is planned that the LUNA-GLOB spacecraft will deliver an orbiter and 14 landers to the Moon.The schematic diagram of the spacecraft flight to the Moon is shown in Fig. 1

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Fig. 1. Schematic diagram of the LUNA-GLOB SC flight to the Moon.

. The flight time is estimated at 4.5 days. Some 33 h before the spacecraft approaches the Moon a container with 10 small high-velocity penetrators (SHVP) will be separated from the spacecraft. It will continue the flight to the Moon autonomously. When the container is at a close distance to the Moon it will intensively rotate, and the penetrators will be separated from it. They will continue their flight to the Moon, and at a rate of 2.6 km⧸s, they will penetrate the surface.Then, two large penetrators (LP) will separate and continue the flight autonomously. In due time they will decelerate and penetrate the surface at a rate of 80–100 m⧸s.Finally, after leaving the satellite orbit the polar station (PS) will land to the South Polar Region at a rate of 80–100 m⧸s.All 12 penetrators, which will be dropped from the spacecraft, have seismometers. They are intended for the research into the internal structure of the Moon that is one of the main scientific objectives of the project. The (PS) accommodates a complex of instruments intended for the solution of another objective of the project: the search for volatiles.     

Earth satellites in resonance with the Moon and the Sun as objects of laser ranging. Analytical solution for their motion

Distant Earth satellites repeating nearly periodically their configurations both with the Moon and the Sun may appear to be even more convenient carriers of laser retroreflectors than the Moon, when geodetic applications are of primary interest. The analytical solution for the motion of a satellite in resonance both with the Moon and the Sun has been outlined in this paper, the periodic orbit of the planar restricted four-body problem being taken as an intermediary. The Von Zeipel transformation gives the Hamiltonian not depending on the fast variables. The stationary solution for this Hamiltonian has been found. Then the non-homogeneous variation equations have been formed, taking into account the orbital eccentricities of the Moon and the Sun. The solution of these equations has been obtained and its accuracy has been tested by numerical integration.Presented at the XXII Congress of the International Astronautical Federation, Brussels, Belgium September 20–25, 1971.     

Extension of Cassini's Laws

We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc.     

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.     

Comments on the figure of the moon based on preliminary results from laser altimetry

Range measurements from the orbiting spacecraft to the lunar surface were made during the Apollo 15 mission using a laser altimeter. The measurements were made in a plane inclined at approximately 26° with respect to the lunar equator. Analysis of measurements made during one complete lunar revolution indicates that the figure of the Moon is very complex. The lunar far side appears to be considerably rougher than the near side in this plane. There appears to be a very large depression on the far side centered at approximately 180° longitude. The near-side maria are depressed with respect to surrounding terrae. These data provide some proof that there is a displacement between the center of figure and the center of mass of the Moon.     

Fission and geosynchronous release of the Moon

The problem of the origin of the Moon has led to various hypotheses: simultaneous accretion, fission, capture, etc. These theories were based primarily on global mechanical considerations. New geological data (Turcotteet al., 1974; Kahn and Pompea, 1978) have led to fresh approaches and new versions of these theories.As suggested by Wise (1969) and O'Keefe (1972), the initial Earth may have taken unstable forms when radial segregation sped up the rotation. The Moon may have been created as the small part of the pyroid of Poincaré.Fission theory was mainly discarded, in the past, on the basis of energy considerations. We are now arriving at the conclusion that these considerations are void if the fission was followed by a very long period of geostationary rotation of the Moon at a distance of about 3 Earth radius (i.e., out of the Roche limit). Indeed the large amount of energy of the initial system could have been released slowly and therefore evacuated by losses of material and radiation.The accretion of the Earth and the radial segregation of heavy chemicals toward the center has led to a differential rotation of the different layers with a faster rotation at the center. During the geostationary period the Moon was synchronous with respect to the surface layer. That Earth-Moon system has both a correct angular momentum and a large stability provided that the viscosity of intermediate layers was small enough, which is in concordance with its high temperature.Even with a very hot system, a superficial cold layer appears because of its low conductivity and the radiation equilibrium with outer space. This implies a slow loss of energy: the geosynchronous Moon receded extremely slowly.During the geostationary period lithophile elements were extracted with water by the radial segregation and were deposited in the area facing the Moon. One massive continent was formed, as suggested by Grjebine (1978).As the continent became thicker and sank into the mantle, convection currents appeared and speeded up the cooling of the Earth. The viscosity increased and the synchronization between the Moon and the surface of the Earth became more difficult to maintain. When synchronism was broken important lunar tides transferred energy and momentum from the Earth to the Moon which receded toward its present position and the modification of its equilibrium shape explains the formation of lunar maria in the near side.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.     

The origin of the Moon: Theories involving joint formation with the Earth

A planet the size of the Earth or the Moon is much like a blast furnace; it produces slag-like rock floating on a mass of liquid metal. In the Earth, the mantle and crust are the slag, and the core is the liquid iron.In the Moon, there is clear chemical evidence that liquid iron was separated from the mass, but the Moon has no detectable iron core. This points to some kind of joint origin, which put the metallic iron in the Earth's core. For instance, the Moon might have been a detached part of the rocky matter of the Earth, as suggested by G. H. Darwin in the 1880's. But is is also clear, as Ringwood has pointed out, the there has been an enormous loss of volatiles from both Earth and Moon, but especially from the Moon. It may be that the Moon formed from a sediment-ring of small bodies detached somehow from the outer parts of the Earth, as Öpik has suggested.If tektites come from the Moon, then Darwin's suggestion is probably right; if they come from the Earth, then the Öpik-Ringwood sediment ring may be the origin.Paper presented at the AAAS Symposium on the Early History of the Earth and Moon in Philadelphia on 28 December 1971.     

The formation of the moon

Supporting evidence for the fission hypothesis for the origin of the Moon is offered. The maximum allowable amount of free iron now present in the Moon would not suffice to extract the siderophiles from the lunar silicates with the observed efficiency. Hence extraction must have been done with a larger amount of iron, as in the mantle of the Earth, of which the Moon was once a part, according to the fission hypothesis. The fission hypothesis gives a good resolution of the tektite paradox. Tektites are chemically much like products of the mantle of the Earth; but no physically possible way has been found to explain their production from the Earth itself. Perhaps they are a product of late, deep-seated lunar volcanism. If so, the Moon must have inside it some material with a strong resemblance to the Earth's mantle. Two dynamical objections to fission are shown to be surmountable under certain apparently plausible conditions.