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.     

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.     

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.     

Lunar shape does not record a past eccentric orbit

The Moon has long been known to have an overall shape not consistent with expected past tidal forces. It has recently been suggested (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) that the present lunar moments of inertia indicate a past high-eccentricity orbit and, possibly, a past non-synchronous spin-orbit resonance. Here I show that the match between the lunar shape and the proposed orbital and spin states is much less conclusive than initially proposed. Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) spin and shape evolution scenarios also completely ignore the physics of the capture into such resonances, which require prior permanent deformation, as well as tidal despinning to the relevant resonance. If the early lunar orbit was eccentric, the Moon would have been rotating at an equilibrium non-synchronous rate determined by it eccentricity. This equilibrium supersynchronous rotation would be much too fast to allow a synchronous spin-orbit lock at e = 0.49, while the capture into the 3:2 resonance is possible only for a very constrained lunar eccentricity history and assuming some early permanent lunar tri-axiality. Here I show that large impacts in the early history of the Moon would have frequently disrupted this putative resonant rotation, making the rotation and eccentricity solutions of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) unstable. I conclude that the present lunar shape cannot be used to support the hypothesis of an early eccentric lunar orbit.     

Coupled thermal-orbital evolution of the early Moon

Coupled thermal-orbital histories of early lunar evolution are considered in a simple model. We consider a plagioclase lid, overlying a magma ocean, overlying a solid mantle. Tidal dissipation occurs in the plagioclase lid and heat transport is by conduction and melt migration. We find that large orbital eccentricities can be obtained in this model. We discuss possible consequences of this phase of large eccentricities for the shape of the Moon and geochronology of lunar samples. We find that the orbit can pass through the shape solution of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M. [2006]. Science 313, 652), but we argue that the shape cannot be maintained against elastic deformation as the orbit continues to evolve.     

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.     

Mercury and Moon He exospheres: Analysis and modeling

Helium is one of the first elements clearly identified in the lunar exosphere (Hoffman, J.H., Hodges, R.R., Johnson, F.S., Evans, D.E. [1973]. Proc. Lunar Sci. Conf. 3, 2865–2875). Apollo 17 measured the He density at the surface during four lunations. It confirmed the expected day to night asymmetry of the He exosphere with a maximum density near the dawn terminator on the nightside. Few years later, the first detection of Mercury’s He exosphere was successfully obtained by Mariner 10 (Broadfoot, A.L., Shemansky, D.E., Kumar, S. [1976]. Geophys. Res. Lett. 3, 577–580). These observations highlighted similar global distribution of the He exosphere at Mercury and at the Moon, but also significant differences that have never been convincingly explained.In this paper, we model the He exosphere at the Moon and Mercury with the same approach. The energy accommodation of the exospheric He particles interacting with the surface can be roughly constrained using Apollo 17 and Mariner 10 measurements. Neither a low energy accommodation, as suggested by Shemansky and Broadfoot (Shemansky, D.E., Broadfoot, A.L. [1977]. Rev. Geophys. 15, 491–499), nor a full energy accommodation, as suggested by Hodges (Hodges Jr., R.R. [1975]. The Moon, 14, 139–157), can fit all the observations. These observations and their modeling suggest a diurnal variation of the energy distribution of the He ejected from the surface that cannot be explained satisfactorily by any of the present theories on the gas–surface interaction in surface-bounded exospheres.     

The fate of primordial lunar Trojans

Multiple large impact basins on the lunar nearside formed in a relatively-short interval around 3.8-3.9 Gyr ago, in what is known as the Lunar Cataclysm (LC; also known as Late Heavy Bombardment). It is widely thought that this impact bombardment has affected the whole Solar System or at least all the inner planets. But with non-lunar evidence for the cataclysm being relatively weak, a geocentric cause of the Lunar Cataclysm cannot yet be completely ruled out [Ryder, G., 1990. Eos 71, 313, 322-323]. In principle, late destabilization of an additional Earth satellite could result in its tidal disruption during a close lunar encounter (cf. [Asphaug, E., Agnor, C.B., Williams, Q., 2006. Nature 439, 155-160]). If the lost satellite had D>500 km, the resulting debris can form multiple impact basins in a relatively short time, possibly explaining the LC. Canup et al. [Canup, R.M., Levison, H.F., Stewart, G.R., 1999. Astron. J. 117, 603-620] have shown that any additional satellites of Earth formed together with (and external to) the Moon would be unable to survive the rapid initial tidally-driven expansion of lunar orbit. Here we explore the fate of objects trapped in the lunar Trojan points, and find that small lunar Trojans can survive the Moon's orbital evolution until they and the Moon reach 38 Earth radii, at which point they are destabilized by a strong solar resonance. However, the dynamics of Trojans containing enough mass to cause the LC (diameters >150 km) is more complex; we find that such objects do not survive the passage through a weaker solar resonance at 27 Earth radii. This distance was very likely reached by the Moon long before the LC, which seems to rule out the disruption of lunar Trojans as a cause of the LC.     

Thermal model of water and CO activity of Comet C/1995 O1 (Hale-Bopp)

An investigation of the activity of Comet C/1995 O1 (Hale-Bopp) with a thermophysical nucleus model that does not rely on the existence of amorphous ice is presented. Our approach incorporates recent observations allowing to constrain important parameters that control cometary activity. The model accounts for heat conduction, heat advection, gas diffusion, sublimation, and condensation in a porous ice-dust matrix with moving boundaries. Erosion due to surface sublimation of water ice leads to a moving boundary. The movement of the boundary is modeled by applying a temperature remapping technique which allows us to account for the loss in the internal energy of the eroded surface material. These kind of problems are commonly referred to as Stefan problems. The model takes into account the diurnal rotation of the nucleus and seasonal effects due to the strong obliquity of Hale-Bopp as reported by Jorda et al. (Jorda, L., Rembor, K., Lecacheux, J., Colom, P., Colas, F., Frappa, E., Lara, L.M. [1997]. Earth Moon Planets 77, 167-180). Only bulk sublimation of water and CO ice are considered without further assumptions such as amorphous ices with certain amount of occluded CO gas. Confined and localized activity patterns are investigated following the reports of Lederer and Campins (Lederer, S.M., Campins, H. [2002]. Earth Moon Planets 90, 381-389) about the chemical heterogeneity of Hale-Bopp and of Bockelée-Morvan et al. (Bockelée-Morvan, D., Henry, F., Biver, N., Boissier, J., Colom, P., Crovisier, J., Despois, D., Moreno, R., Wink, J. [2009]. Astron. Astrophys. 505, 825-843) about a strong CO source at a latitude of 20°. The best fit to the observations of Biver et al. (Biver, N. et al. [2002]. Earth Moon Planets 90, 5-14) is obtained with a low thermal conductivity of 0.01 W m⁻¹ K⁻¹. This is in agreement with recent results of the Deep Impact mission to 9P/Tempel 1 (Groussin, O., A’Hearn, M.F., Li, J.-Y., Thomas, P.C., Sunshine, J.M., Lisse, C.M., Meech, K.J., Farnham, T.L., Feaga, L.M., Delamere, W.A. [2007]. Icarus 187, 16-25) and with previous thermal simulations (Kührt, E. [1999]. Space Sci. Rev. 90, 75-82). The water production curve matches the production rates well from −4 AU pre-perihelion to the outgoing leg while the model does not reproduce so well the water production beyond 4 AU pre-perihelion. The CO production curve is a good fit to the measurements of Biver et al. (2002) over the whole measured heliocentric range from −7 AU pre- to 15 AU post-perihelion.     

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.     

Calibration against the Moon I: A disk-resolved lunar model for absolute reflectance calibration

We present a model of the absolute radiance of the disk-resolved Moon at visible to near infrared wavelengths. It has been developed in order to use the Moon as a calibration reference, particularly by space-based sensors observing the Earth. We begin with the development of Hillier et al. (Hillier, J., Buratti, B., Hill, K. [1999]. Icarus 141, 205-225) for the reflectance as a function of phase angle and base the lunar reflectance on the Clementine 0.750 μm basemap. We adopt Hapke’s (Hapke, B. [2002]. Icarus 157, 523-534) expression for the multiple scattering term, including the more accurate approximation to the Chandrasekhar H function. The geometry is based on the Jet Propulsion Laboratory Lunar Ephemeris DE 421, and the topographic slope is from the Kaguya-LALT laser altimetry (Araki, H., and 10 colleagues [2009]. Science 323, 897-900). We define three types of terrain by combining the reflectance from the Clementine basemap and the topographic model to specify maria, highlands, and crater regions, and allow mixed types between each class. Parameters of the model are solved for as a function of surface type and wavelength by comparison against data “chips” from the Robotic Lunar Observatory (ROLO; Kieffer, H.H., Stone, T.C. [2005]. Astron. J. 129, 2887-2901). The reflectance in any waveband may be computed by spectral interpolation of the model predictions relative to the scaled Apollo 16 soil spectrum. The accuracy of the model, evaluated against ROLO imagery, was found to be 2-4%.     

Resolved spectroscopy of Mercury in the near-IR with SpeX/IRTF

We present resolved near-infrared spectra of Mercury scanning 70% of the surface in latitude and longitude from three separate observations, allowing us to perform a compositional investigation of its surface. By scanning the surface we find that all spectra in our sample are remarkably similar suggesting overall compositional homogeneity. We do, however, observe a slope difference between the spectra. These slope changes are most likely due to differences in the emission angle over different parts of the surface. We confirm the presence of a 1.1 μm feature that had been previously detected (Warell, J. et al. [2006]. Icarus 180, 281-291) and attributed to Ca-rich clinopyroxene. Finally, we investigated Mercury’s surface composition by comparing its spectrum with ground-based lunar spectra, lunar soil spectra collected in the laboratory, and analysis with a simple linear mixing model using various minerals as end-members. The result of this compositional investigation reveals that Mercury’s surface composition is likely to be quite different from the Moon’s. While low-Ca iron-rich pyroxenes are main surface components on the Moon (abundance varying from ∼5% to ∼35%), their abundance on Mercury may not exceed 5%. We also find that a Ca-rich clinopyroxene (in the hedenbergite-diopside series) is likely to be a main component of Mercury’s surface whereas this mineral is almost absent on the Moon. Our analysis also suggests the possible presence of olivine. We find that Mercury’s slope is less red than that of the Moon, in agreement with results from MESSENGER (McClintock, W.E., and 12 colleagues [2008]. Science 321, 62-65), and composition rather than variation of space weathering is likely the cause of this difference.     

On volatilization in vacuum of the elements from the planet surface soil melts

Endogeneous and exogeneous events resulting in the appearance of the large volumes of melted substance on the lunar surface should be accompanied by volatilization of some elements from melts in vacuum [1]. However, it is not clear up to now whether volatilization on the Moon has indeed occurred and how the lower (in comparison with the Earth) content of some elements (Na, K and others) in lunar soil can be explained. There are contradictory opinions on these problems in publications (O'Hara, Ringwood and others) [2]. The numerous laboratory investigations of the similar processes are insufficient [3–15; 36] for interpretation of the results for the lack of an adequate physical model and the theory of these processes and also due to the narrow range of the parameters used (T, p, τ) and the experimental regimes. In the present paper a physical model is developed, which is based on experimental data; together with a theory of the process of volatilization of the volatile components of rock melts in vacuum, taking into account an adsorption of the residual atmosphere gases; all these allow us to interpret such processes successfully. As a result, some preliminary conclusions have been drawn about such phenomena on the Moon and their laboratory simulation.     

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.     

The effect of surface topography on the lunar photoelectron sheath and electrostatic dust transport

The dayside near-surface lunar plasma environment is electrostatically complex, due to the interaction between solar UV-induced photoemission, the collection of ambient ions and electrons, and the presence of micron and sub-micron sized dust grains. Further complicating this environment, although less well understood in effect, is the presence of surface relief, typically in the form of craters and/or boulders. It has been suggested that such non-trivial surface topography can lead to complex electrostatic potentials and fields, including “mini-wakes” behind small obstacles to the solar wind flow and “supercharging” near sunlit-shadowed boundaries (Criswell, D.R., De, B.R. [1977]. J. Geophys. Res. 82 (7); De, B.R., Criswell, D.R. [1977]. J. Geophys. Res. 82 (7); Farrell, W.M., Stubbs, T.J., Vondrak, R.R., Delory, G.T., Halekas, J.S. [2007]. Geophys. Res. Lett. 34; Wang, X., Horányi, M., Sternovsky, Z., Robertson, S., Morfill, G.E. [2007]. Geophys. Res. Lett. 34, L16104). In this paper, we present results from a three-dimensional, self-consistent, electrostatic particle-in-cell code used to model the dayside near-surface lunar plasma environment over a variety of local times with the presence of a crater. Additionally, we use the particle-in-cell model output to study the effect of surface topography on the dynamics of electrostatic dust transport, with the goal of understanding previous observations of dust dynamics on the Moon and dust ponding on various asteroids.     

The kinematical mechanism for the perturbation of the rotational axis in the rotation of the elastic Earth

Kubo (Celest Mech Dyn Astron 110:143–168, 2011) investigated the kinematical structure of the perturbation in the rotation of the elastic Earth due to the deformation caused by the outer bodies. In that paper, while the mechanism for the perturbation of the figure axis was made clear, that for the rotational axis was not shown explicitly. In the present study, following the same method, the structure of the perturbation of the rotational axis is investigated. This perturbation consists of the direct perturbation and the convective perturbation. First the direct perturbation is shown to be (A − C)/A times as large as that of the figure axis, coinciding with the analytical expressions obtained in preceding studies by other authors. As for the convective perturbation, which appears only in the perturbation of the rotational axis but not in that of the figure axis, it is shown to be (A − C)/A times the angular separation between the original figure axis and the induced figure axis produced by the elastic deformation, A and C being the principal moments of inertia of the Earth. If the perturbing bodies are motionless, the conclusion of Kubo (Celest Mech Dyn Astron 105:261–274, 2009) holds strictly, i.e. the sum of the direct and the convective perturbations of the rotational axis coincides with the perturbation of the figure axis.     

Reassessing the origin of Triton

The origin of Neptune’s large, circular but retrograde satellite Triton has remained largely unexplained. There is an apparent consensus that its origin lies in it being captured, but until recently no successful capture mechanism has been found. Agnor and Hamilton (Agnor, C.B., Hamilton, D.P. [2006]. Nature 441, 192-194) demonstrated that the disruption of a trans-neptunian binary object which had Triton as a member, and which underwent a very close encounter with Neptune, was an effective mechanism to capture Triton while its former partner continued on a hyperbolic orbit. The subsequent evolution of Triton’s post-capture orbit to its current one could have proceeded through gravitational tides (Correia, A.C.M. [2009]. Astrophys. J. 704, L1-L4), during which time Triton was most likely semi-molten (McKinnon, W.B. [1984]. Nature 311, 355-358). However, to date, no study has been performed that considered both the capture and the subsequent tidal evolution. Thus it is attempted here with the use of numerical simulations. The study by Agnor and Hamilton (Agnor, C.B., Hamilton, D.P. [2006]. Nature 441, 192-194) is repeated in the framework of the Nice model (Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F. [2005]. Nature 435, 459-461) to determine the post-capture orbit of Triton. After capture Triton is then subjected to tidal evolution using the model of Mignard (Mignard, F. [1979]. Moon Planets 20, 301-315; Mignard, F. [1980]. Moon Planets 23, 185-201). The perturbations from the Sun and the figure of Neptune are included. The perturbations from the Sun acting on Triton just after its capture cause it to spend a long time in its high-eccentricity phase, usually of the order of 10 Myr, while the typical time to circularise to its current orbit is some 200 Myr, consistent with earlier studies. The current orbit of Triton is consistent with an origin through binary capture and tidal evolution, even though the model prefers Triton to be closer to Neptune than it is today. The probability of capturing Triton in this manner is approximately 0.7%. Since the capture of Triton was at most a 50% event - since only Neptune has one, but Uranus does not - we deduce that in the primordial trans-neptunian disc there were some 100 binaries with at least one Triton-sized member. Morbidelli et al. (Morbidelli, A., Levison, H.F., Bottke, W.F., Dones, L., Nesvorný, D. [2009]. Icarus 202, 310-315) concludes there were some 1000 Triton-sized bodies in the trans-neptunian proto-planetary disc, so the primordial binary fraction with at least one Triton-sized member is 10%. This value is consistent with theoretical predictions, but at the low end. If Triton was captured at the same time as Neptune’s irregular satellites, the far majority of these, including Nereid, would be lost. This suggests either that Triton was captured on an orbit with a small semi-major axisa ? 50R_N (a rare event), or that it was captured before the dynamical instability of the Nice model, or that some other mechanism was at play. The issue of keeping the irregular satellites remains unresolved.     

Comment on “Constraints on the source of lunar cataclysm impactors” (Cuk et al., 2010, Icarus 207, 590-594)

Cuk et al. (Cuk, M., Gladman, B.J., Stewart, S.T. [2010]. Icarus 207, 590-594) argue that the projectiles bombarding the Moon at the time of the so-called lunar cataclysm could not have been mainbelt asteroids ejected by purely gravitational means, in contradiction with a conclusion that was reached by Strom et al. (Strom, R.G., Malhotra, R., Ito, T., Yoshida, F., Kring, D.A. [2005]. Science 309, 1847-1850). We demonstrate that Cuk et al.’s argument is erroneous because, contrary to their arguments, the lunar highlands do register the cataclysm impacts, lunar class 1 craters do not represent the size distribution of the cataclysm craters, and the crater size distributions on the late-forming basins are quite similar to those of the highlands craters, albeit at a lower number density due to the rapid decline of the impact flux during the cataclysm.     

Understanding measured water rotational temperatures and column densities in the very innermost coma of Comet 73P/Schwassmann–Wachmann 3 B

Direct sublimation of a comet nucleus surface is usually considered to be the main source of gas in the coma of a comet. However, evidence from a number of comets including the recent spectacular images of Comet 103P/Hartley 2 by the EPOXI mission indicates that the nucleus alone may not be responsible for all, or possibly at times even most, of the total amount of gas seen in the coma. Indeed, the sublimation of icy grains, which have been injected into the coma, appears to constitute an important source. We use the fully-kinetic Direct Simulation Monte Carlo model of Tenishev et al. (Tenishev, V.M., Combi, M.R., Davidsson, B. [2008]. Astrophys. J., 685, 659?677; Tenishev, V.M., Combi, M.R., Rubin, M. [2011]. Astrophys. J., 732) to reproduce the measurements of column density and rotational temperature of water in Comet 73P-B/Schwassmann–Wachmann 3 obtained with a very high spatial resolution of ～30 km using IRCS/Subaru in May 2006 (Bonev, B.P., Mumma, M.J., Kawakita, H., Kobayashi, H., Villanueva, G.L. [2008]. Icarus, 196, 241?248). For gas released solely from the cometary nucleus at a heliocentric distance of 1 AU, modeled rotational temperatures start at 110 K close to the surface and decrease to only several tens of degrees by 10–20 nucleus radii. However, the measured decay of both rotational temperature and column density with distance from the nucleus is much slower than predicted by this simple model. The addition of a substantial (distributed) source of gas from icy grains in the model slows the decay in rotational temperature and provides a more gradual drop in column density profiles. Together with a contribution of rotational heating of water molecules by electrons, the combined effects allow a much better match to the IRCS/Subaru observations. From the spatial distributions of water abundance and temperature measured in 73P/SW3-B, we have identified and quantified multiple mechanisms of release. The application of this tool to other comets may permit such studies over a range of heliocentric and geocentric distances.