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.     

月球卫星轨道变化的分析解

由于月球自转缓慢及其引力位的特点,使得讨论月球卫星与人造地球卫星轨道变化的方法有所不同。     

Selenographic control

Evaluation of selenographic data obtained with use of different observational means require the formulation of rigorous algorithms connecting the systems of coordinates, which the various methods have been referred to. The lunar principal axes of inertia are suggested as most appropriate for reference in lunar mapping and selenographic coordinate catalogues. The connection between the instantaneous axis of lunar rotation (involved in laser ranging, radar studies, astronomical observations from the surface of the Moon and VLBI observations of ALSEPs), the ecliptic system of coordinates (which in reductions of observations was considered as fixed in space), the Cassini mean selenographic coordinates (to which physical libration measures were referred), the lunar principal axes of inertia and the invariable plane of the solar system is discussed.On leave from the University of Manchester, England.Lunar Science Institute Contribution No. 138.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.     

月球物理天平动对环月轨道器运动的影响

月球物理天平动是月球赤道在空间真实的摆动，会导致月球引力场在空间坐标系中的变化，从而引起环月轨道器(以下称为月球卫星)的轨道变化，这与地球的岁差章动现象对地球卫星轨道的影响类似．采用类似对地球岁差章动的处理方法，讨论月球物理天平动对月球卫星轨道的影响，给出相应的引力位的变化及卫星轨道的摄动解，清楚地表明了月球卫星轨道的变化规律，并和数值解进行了比对，从定性和定量方面作一讨论．     

Analysis of Stability of Orbits of Artificial Lunar Satellites and Configuring of a Lunar Satellite Navigation System

The analysis of the Moon artificial satellite orbits stability and satellite system configuring are important issues of lunar orbital navigational system development. The article analyses the influence of different combinations of perturbations on Moon artificial satellite’s obits evolution. The method of Moon artificial satellite’s orbital evolution analysis is offered; general stability regions of Moon artificial satellite’s orbits are defined and the quality characteristics of the selected orbital groups of the satellite system are evaluated.     

Computer simulation of observations of stars from the moon using the polar zenith telescope of the Japanese project ILOM

The paper briefly describes the purpose and features of the Japanese project ILOM (In-situ Lunar Orientation Measurement) in which it is planned to install the zenith telescope with a CCD lens on one of the poles of the Moon for the observation of stars in order to determine the physical libration of the Moon (PhLM). The studies presented in this paper are the result of the first stage of the theoretical support of the project:

1. The compilation of the list of stars within the field of view of the telescope during the precessional motion of the lunar pole.
2. Modeling and analysis of the behavior of stellar tracks during the observation period.
3. Simulation and testing of the sensitivity of the measured selenographic star coordinates to changes in the parameters of the dynamic model of the Moon and the elastic parameters of the lunar body.

Direct and inverse PhLM problems are discussed. Within the scope of the direct problem visible “daily parallels” and one-year star tracks are calculated. Their behavioral features when observed from the lunar surface are shown. At this stage of the simulation selenographic star coordinates for the four models of the gravitational field of the Moon have been compared, i.e., the model constructed on the basis of the lunar laser ranging (LLR), GLGM-2, LP150Q, and SGM100h. It is shown that even when comparing modern models LP150Q and SGM100h stellar tracks differ from the arc by more than 10 ms of arc. At the stage of the inverse problem, the manifestation of viscoelastic properties of the Moon in selenographic coordinates has been studied. In the spectrum of the simulated residual differences harmonics have been identified which can serve as indicators to refine parameters, Love number k ₂ and the delay time characterizing the viscous properties of the lunar body.     

月球卫星轨道力学综述

月球探测器的运动通常可分为3个阶段，这3个阶段分别对应3种不同类型的轨道：近地停泊轨道、向月飞行的过渡轨道与环月飞行的月球卫星轨道。近地停泊轨道实为一种地球卫星轨道；过渡轨道则涉及不同的过渡方式(大推力或小推力等)；环月飞行的月球卫星轨道则与地球卫星轨道有很多不同之处，它决不是地球卫星轨道的简单克隆。针对这一点，全面阐述月球卫星的轨道力学问题，特别是环月飞行中的一些热点问题，如轨道摄动解的构造、近月点高度的下降及其涉及的卫星轨道寿命、各种特殊卫星(如太阳同步卫星和冻结轨道卫星等)的轨道特征、月球卫星定轨等。     

Some orbital characteristics of lunar artificial satellites

In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.     

Determination of some physical parameters of the Moon with Lunar Laser Ranging data

Information about the structure of lunar interior and evolution could be obtained from measurements of lunar free librations, gravitational field, dissipation etc.. In this paper the precision of determining free librations with Lunar Laser Ranging (LLR) data are estimated. Using the observing data from four telescopes for eighteen years the amplitudes and phases of free librations, the moments of inertia ratio of The Moon were determined.     

On the perturbations of a close-Earth satellite due to lunar inequalities

The lunar disturbing function for a close-Earth satellite is expressed as a sum of products of harmonics of the satellite's position and harmonics of the Moon's position, and the latter are expanded about a rotating and precessing elliptic orbit inclined to the ecliptic. The deviations of the Moon from this approximate orbit are computed from Brown's lunar theory andthe perturbations in satellite orbital elements due to these inequalities are derived. Numerical calculations indicate that several perturbations in the position of the satellite's node and perigee have magnitudes on the order of one meter.The author is supported in part by a National Science Foundation Graduate Fellowship.     

Lunar libration tables

The selenographic direction cosines of the Earth and of the pole of the ecliptic are developed in trigonometric series; the time dependence of each term enters through the Delaunay arguments. These and other series are tabulated for different libration parameters to provide means for calculating at any epoch the lunar librations and their partial derivatives with respect to the parameters.     

Estimation of solar illumination on the Moon: A theoretical model

The solar illumination conditions on the lunar surface represent a key resource with respect to returning to the Moon. As a supplement to mapping the solar illumination by exploring data, lighting simulations using high-resolution topography could produce quantitative illumination maps. In this study, a theoretical model is proposed for estimating the solar illumination conditions. It depends only on the solar altitude and topographical factors. Besides the selenographic longitude and latitude, the former is determined by the selenographic longitude and latitude at the subsolar site, the geocentric ecliptical latitude, and the dimensionless distance of the Sun–Moon relative to 1 AU, which are function of time. The latter is determined by comparing the elevations in solar irradiance direction within 210 km in which the topography might shadow the behind sites to the critical elevations determining whether the behind sites are shadowed or not. Compared to Zuber's model, the model proposed in this study is simpler and easier for computing. It is parameterized with selenographic coordinates, elevations, and time. With high-resolution topography data, the solar illumination conditions at any selenographic coordination could be estimated by this model at any date and time. The lunar surface is illuminated when the solar altitude is non-zero and all the elevations within 210 km in solar irradiance direction are lower than the critical elevations. Otherwise it would be shadowed.     

Insight-building models for lunar range and range rate

The analysis of range or Doppler data between sites on the Earth and Moon requires an accurate computation of the lunar orbit and detailed models of the orientation of the Earth and Moon. Models constructed to understand range and range rate can lack detail, but if they include the largest lunar orbit variations, tracking stations on a rotating Earth, and lunar sites on a synchronously rotating Moon, then they will display the largest effects for orbit elements, Earth orientation, tracking station locations, and lunar site coordinates. The range and range rate are expanded into periodic series. To understand accurate solutions, the largest periodic terms that are sensitive to various solution parameters indicate the sensitivity of data to solution parameters and the time spans needed for their determination. Conclusions include: cylindrical coordinates work well for sites on the rapidly rotating Earth, but Cartesian coordinates are more natural for the synchronously rotating Moon since the series for the three coordinate projections are distinct. For range and range rate data, daily, semimonthly, monthly, and longer periods are present. For Doppler data, the daily periods may be stronger and more useful than the long periods, particularly for terms associated with the terrestrial tracking station. Doppler data do not determine the lander coordinate toward the Earth well. Observational strategies for range and Doppler data are not identical. For all data types, one wishes a variety of hour angles, lunar declinations, times of month, and longer periods. A long span of high-quality range data can improve the lunar orbit, orientation of the Earth’s equator, and physical librations. The locations of new lunar sites or new tracking stations can be determined from shorter spans of data.     

On the systems of selenographic coordinates,their determination and terminology

Attention is drawn to the absence in literature of the precise definitions of selenographic and celestial selenocentric coordinate systems. In certain cases inaccuracies in the formulation of the first Cassini law occur. This is due to the fact that the principal directions dealt with in the theory of lunar rotation are being constantly confused. A clear-cut definition of the principal coordinate systems concerned with the lunar rotation is given. It is indicated that there is no necessity in a special astronomical time service on the Moon. Since the future expeditions to the Moon will be able to keep terrestrial time, the problem of the hour angle is simply solved by the Formula (11).     

Physical librations due to the third and fourth degree harmonics of the lunar gravity potential

The existence of third and fourth harmonics of the lunar gravity potential gives rise to sizable lunar physical librations. Using one recent set of potential estimates, the following effects are noted: the mean sub-Earth point is displaced from the earthward principal moment of inertia axis by 168″; the inclination of the lunar equator to the ecliptic is decreased by 14″.5; and a six year period libration in longitude, with amplitude 13″.1, is induced.     

Effects of a physical librations of the moon caused by a liquid core,and determination of the fourth mode of a free libration

An analytical theory of lunar physical librations based on its two-layer model consisting of a non-spherical solid mantle and ellipsoidal liquid core is developed. The Moon moves on a high-precision orbit in the gravitational field of the Earth and other celestial bodies. The defined fourth mode of a free libration is caused by the influence of the liquid core, with a long period of 205.7 yr, with amplitude S = 0″0395 and with an initial phase Π₀ = ?134° (for the initial epoch 2000.0). Estimates of dynamic (meridional) oblatenesses of a liquid core of the Moon have been estimated: ?_D = 4.42 × 10^?4, μ_D = 2.83 × 10^?4 (?_D + μ_D = 7.24 × 10^?4). These results have been obtained as a result of comparison of the developed analytical theory of physical librations of the Moon with the empirical theory of librations of the Moon constructed on the basis of laser observations.     

On the analytic lunar and solar perturbations of a near Earth satellite

The disturbing function of the Moon (Sun) is expanded as a sum of products of two harmonic functions, one depending on the position of the satellite and the other on the position of the Moon (Sun). The harmonic functions depending on the position of the perturbing body are developed into trigonometric series with the ecliptic elementsl, l′, F, D and Γ of the lunar theory which are nearly linear with respect to time. Perturbation of elements are in the form of trigonometric series with the ecliptic lunar elements and the equatorial elements ω and Ω of the satellite so that analytic integration is simple and the results accurate over a long period of time.     

Lunar perturbations of artificial satellites of the earth

Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.     

Luni-solar perturbations of an Earth satellite

Luni-solar perturbations of the orbit of an artificial Earth satellite are given by modifying the analytical theory of an artificial lunar satellite derived by the author in recent papers. Expressions for the first-order changes, both secular and periodic, in the elements of the geocentric Keplerian orbit of the earth satellite are given, the moon's geocentric orbit, including solar perturbations in it, being found by using Brown's lunar theory.The effects of Sun and Moon on the satellite orbit are described to a high order of accuracy so that the theory may be used for distant earth satellites.     

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.