Secular tidal changes in lunar orbit and Earth rotation

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.     

Simulations of multimode bolometric interferometers

We describe a procedure for the numerical modelling of astronomical interferometers, with particular relevance to far-infrared and submillimetre wavelengths. The scheme is based on identifying a set of modes that carry power from the sky to the detector. The procedure is extremely general, and can be used to model scalar or vector fields, in any state of coherence and polarization, the only limitation being that the propagation of a coherent field through the system be described by an integral transform, a constraint that is in practise always met.
We present simulations of ideal, multimode two-dimensional interferometers, and show that the modal theory reproduces the correct behaviour of both Michelson and Fizeau interferometers. We calculate simulated visibility data for a multimode bolometric Michelson interferometer, with a synthesized source, and produce a dirty map, recovering the original source with the usual artefacts associated with interferometers.     

High-resolution radar maps of the lunar surface at 3.8-cm wavelength

The entire earth-facing lunar surface has been mapped at a resolution of 2 km using the 3.8-cm radar of Haystack Observatory. The observations yield the distribution of relative radar backscattering efficiency with an accuracy of about 10% for both the polarized (primarily quasispecular or coherent) and depolarized (diffuse or incoherent) scattered components. The results show a variety of discrete radar features, many of which are correlated with craters or other features of optical photographs. Particular interest, however, attaches to those features with substantially different radio and optical contrasts. An anomaly near 63° is noted in the mean angular scattering law obtained from a summary of the radar data.     

On the evolution of the lunar orbit

It is generally accepted that the Earth-Moon separation is at present increasing due to tidal dissipation. Values for the corresponding lunar deceleration and the related slowing of the Earth's rotation are obtained from astronomical observations and by studies of ancient eclipses. Extrapolation of these values leads to a close approach of the Earth and Moon 1–3 b.y. BP. However, justification for such an extrapolation is required. It has been hypothesized that periodicities in the Precambrian stromatolites can be used to determine the number of solar days in a lunar month prior to 500 m.y. BP. These data combined with dynamic constraints on the number of solar days in a lunar month indicate a close approach of the Earth and Moon at 2.85 ± 0.25 b.y. BP. It is suggested that the mare volcanism on the Moon and high-temperature Archean volcanism on the Earth prior to this date were caused by tidal heating. It is also suggested that the strong tidal heating during a close approach could have contributed to the formation of the first living organisms.     

On the history of the lunar orbit

A frequency-dependent model of tidal friction is used in the determination of the time rate of change of the lunar orbital elements and the angular velocity of the Earth. The variational equations consider eccentricity, the solar tide on the Earth, Earth oblateness, and higher-order terms in the Earth's tidal potential. A linearized solution of the equations governing the precission of the Earth's rotational angular momentum and the lunar ascending node is found. This allows the analytical averaging of the variational equations over the period of relative precession which, though large, is necessarily small in comparison to the time step of the numerical integrator that yields the system history over geological time. Results for this history are presented and are identified as consistent with origin of the Moon by capture. This model may be applied to any planet-satellite system where evolution under tidal friction is of interest.     

Earth albedo and the orbit of Lageos

The long-period perturbations in the orbit of Lageos satellite due to the earth's albedo have been found using a new analytical formalism. The earth is assumed to be a sphere whose surface diffusely reflects sunlight according to Lambert's law. Specular reflection is not considered. The formalism is based on spherical harmonics; it produces equations which hold regardless of whether the terminator is seen by the satellite or not. Specializing in the case of a realistic zonal albedo shows that Lageos' orbital semimajor axis changes periodically by only about a centimeter and the eccentricity by two parts in 10⁵. The longitude of the node increases secularly by about 6×10^–4 arc sec yr^–1. The effect considered here can explain neither the secular decay of 1.1 mm day^–1 in the semimajor axis nor the observed along-track variations in acceleration of order 2×10^–12 ms^–2.     

Dependence of normal albedo of lunar regions on wavelength

Normal albedos of thirteen lunar regions are determined at five wavelengths between 4000 and 8000 Å. The deduced values agree with those of the previous investigators except those given by Gehrelset al. (1964) and Van Diggelen (1965). They increase monotonically with the wavelength. At the shortest wavelength, a slight enhancement is detected for most of the regions observed.     

Photometry of the lunar surface during lunar eclipses

The photometric observations of the lunar surface during lunar eclipses were carried out on four nights between 1972 to 1978, using the 91 cm reflector of the Dodaira Station of the Tokyo Astronomical Observatory. The photometry was performed in B-, V-, and R-colours, and arranged in accordance with the angular distance from the centre of the Earth's shadow. The results do not show any large systematic differences between the four nights, showing no support for Danjon's proposition.     

The evolution of the lunar orbit revisited. I

After recalling the contribution of Halley, J. Kepler, and G. Darwin to our understanding of the secular acceleration of the Moon, we establish a set of differential equations for the variation of the semi-major axis, and the inclination of the Moon on the maximum area plane. These equations are obtained without expanding the disturbing function, due to the tidal bulge, in term of the elliptic elements. The equations thus obtained are simple enough to allow us a qualitative discussion of the solution, followed by a numerical integration.The results obtained show the Moon was in the distant past in a retrograde orbit, approaching the Earth, its inclination increasing towards 90°; once after a closer approach to the Earth, the Moon receeded and it will finally reach an equilibrium point, the orbital and the equatorial planes being blended.The solution of the equations appears as a fascicle of curves, becoming extremely dense as we come nearer to the present. Owing to the high sensitivity of the solution to the initial conditions, a weak disturbance added to our modeled forces may lead to a past situation very different from the conclusion drawn by Goldreich (1966) and MacDonald (1964); the minimal approach distance could be greater than 10 Earth's radii.     

The evolution of the lunar orbit revisited,II

We present here a model for the tidal evolution of an isolated two-body system. Equations are derived, including the dissipation in the planet as in the satellite, in a frequency dependent lag model. The set of differential equations obtained is still valid for large eccentricity, as well as for all inclinations. The reference plane chosen enables us to study the evolution for both the orbital plane and the equatorial plane.The results obtained show the Moon, after having approached the Earth with small variations for the inclination and the eccentricity, exhibits strong increase for the two parameters in the vicinity of the closest approach. In every case the eccentricity tends towards the value 1, whereas the variations of the in clinations are dependent on the magnitude of the dissipation in the satellite.Some qualitative results are also investigated for the final behaviour of satellites such as Triton and the Galilean satellites.     

Simulation of precise orbit determination of lunar orbiters

Based on the ongoing Chinese lunar exploration mission, i.e. the “Chang'e 1” project, precise orbit determination of lunar orbiters is analyzed for the actual geographical distribution and observational accuracy of the Chinese united S-band (USB) observation and control network as well as the very long baseline interferometry (VLBI) tracking network. The observed data are first simulated, then solutions are found after including the effects of various error sources and finally compared. We use the space data analysis software package, GEODYN, developed at Goddard Space Flight Center, NASA, USA. The primary error source of the flight orbiting the moon is the lunar gravity field. Therefore, the (formal) error of JGL165P1, i.e. the model of the lunar gravity field with the highest accuracy at present, is first discussed. After simulating the data of ranging and velocity measurement as well as the VLBI data of the time delay and time delay rate, precise orbit determination is carried out when the error of the lunar gravity field is added in. When the orbit is determined, the method of reduced dynamics is adopted with the selection of appropriate empirical acceleration parameters to absorb the effect of errors in the lunar gravity field on the orbit determination. The results show that for lunar missions like the “Chang'e 1” project, that do not take the lunar gravity field as their main scientific objective, the method of reduced dynamics is a simple and effective means of improving the accuracy of the orbit determination of the lunar orbiters.     

Photometry of the lunar surface

The photometry of the Moon gives us some information about the properties of the lunar surface. The photometric uniformity of the lunar surface as a scattering screen is determined by the shadow phenomena on small irregularities due to the dust layer covering the whole surface. A small component of light (< 10 %) exhibits the features of the luminescence excited by solar radiations.Paper presented to the NATO Advanced Study Institute on Lunar Studies, Patras, Greece, September 1971.     

Thermoluminescence of the lunar surface

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

Application of two special orbits in the orbit determination of lunar satellites

Using inter-satellite range data,the combined autonomous orbit determination problem of a lunar satellite and a probe on some special orbits is studied in this paper.The problem is firstly studied in the circular restricted three-body problem,and then generalized to the real force model of the Earth-Moon system.Two kinds of special orbits are discussed:collinear libration point orbits and distant retrograde orbits.Studies show that the orbit determination accuracy in both cases can reach that of the observations.Some important properties of the system are carefully studied.These findings should be useful in the future engineering implementation of this conceptual study.     

Large-scale evolution of the lunar surface

The first part of the paper describes the relationship between the erosional stage of craters and the crater areal density. It is shown that class-2 and -3 craters are progressively more abundant as the crater areal density increases, while craters of class 4 and 5 are more abundant with decreasing crater areal densities. A geological model is proposed, in which the class of a newly foormed crater is 1. As time progresses, erosional agents will increase the class of the crater to class 2, then 3, and, in some cases, to 4. The length of time between classification steps is not known in terms of years, but is equivalent to the time necessary for the crater density to increase by 2 to 8 craters per unit area for creaters larger than 10 km, and by 10 to 20 for craters larger than 3.5 km. Craters of class 5 and some of class 4 are not formed by the same erosional agents, but are catastrophic, caused either by a mare-producing impact or by flooding of mare material.The second part of the paper presents a method for relatively dating large lunar areas. The method uses the model previously developed. A relative time sequence is constructed using the density of craters of classes 1, 2, and 3 and the percentage of these which is of class 1. As an example, 18 large areas are defined on the lunar near side and are put in temporal order. Mare Serenitatis appears to have the youngest terrain, and an area southwest of the Rupes Altai appears to have the oldest.In the final part of the paper a geological model is developed in order to explain age differences in the terrae. The model calls for rejuvenation of lunar terrains, caused by the seismic waves and ballistic sedimentation resulting from large impacts. The area surrounding Mare Orientale is cited as an example of a terrain so affected. A similar effect on the terrae of the near side could explain the apparent age relationships measured.     

Atlas of lunar radar maps at 70-cm wavelength

The intensity distribution of lunar radar echoes has been mapped for two-thirds of the earth-visible lunar surface at a wavelength of 70 cm. The depolarizing effects of the lunar surface were observed by simultaneously receiving the radar echoes in opposite polarizations. These echoes were mapped with areal resolutions of 25–100 km². Mappings with this resolution confirmed that the young craters have enhanced returns. A few craters were found to have enhanced echoes only from their rims. Backscattering differences were also observed between various areas within a mare, between different highland areas, and between maria and adjacent highlands. These scattering differences were interpreted with a simple model, which assumed that the surface backscattered with varying amounts of quasi-specular and diffuse power. Only an increase in the diffuse power was needed to give the numerical values of the enhancements.     

Observations and analysis of lunar radio emission at 3.09 mm wavelength

Earth, Moon, and Planets - Observations of lunar radio emission were made at 3.09 mm wavelength (97.1 GHz) from April 18 to May 20, 1971. Absolute brightness temperatures were measured for five...     

Dynamical behaviour of Earth orbit crossing asteroids

Computing the maximum and minimum values of the eccentricities and inclinations as functions of the arguments of perihelion for about 7000 numbered asteroids by adopting a simple model it is found that 80 have the minimum perihelion distances less than 1.04 AU. Still, it is proved that 20% of them have no chance of colliding with the Earth, whereas 30 of them have relatively high collision probability as they have orbits similar to those of typical short-period comets.     

Cosmic influences on the early history of the lunar surface

It is pointed out that the observed random distribution of low-angle impact craters over the lunar surface rules out the possibility that particles initially responsible for the origin of such craters had, prior to impact, been in heliocentric orbits. The observed facts are more consistent with a view that particles responsible for most of large primary impact at the earliest stage of lunar history were moving with the Earth-Moon gravitational dipole, and may have represented leftovers from the formation of this pair of cosmic bodies. The application of a similar argument to an equally obvious lack of directional effects in Martian cratering is, however, invalidated by a relatively large inclination of the Martian equator to the orbital plane of this planet.Presented at the NATO Advanced Study Institute on Lunar Studies in Patras, Greece, September 1971.Normally at the Department of Astronomy, University of Manchester, England.