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.     

月球探测器过渡轨道的短弧定轨方法

论述的短弧定轨,是指在无先验信息情况下又避开多变元迭代的初轨计算方法,它需要相应的动力学问题有一能反映短弧内达到一定精度的近似分析解．探测器进入月球引力作用范围后接近月球时可以处理成相对月球的受摄二体问题,而在地球附近,则可处理成相对地球的受摄二体问题,但在整个过渡段的力模型只能处理成一个受摄的限制性三体问题．而限制性三体问题无分析解,即使在月球引力作用范围外,对于大推力脉冲式的过渡方式,相对地球的变化椭圆轨道的偏心率很大(超过Laplace极限),在考虑月球引力摄动时亦无法构造摄动分析解．就此问题,考虑在地球非球形引力(只包含J2项)和月球引力共同作用下,构造了探测器飞抵月球过渡轨道段的时间幂级数解,在此基础上给出一种受摄二体问题意义下的初轨计算方法,经数值验证,定轨方法有效,可供地面测控系统参考．     

An averaging method to study the motion of lunar artificial satellites I: Disturbing Function

We describe a semi-analytical averaging method aimed at the computation of the motion of an artificial satellite of the Moon. In this paper, the first of the two part study, we expand the disturbing function with respect to the small parameters. In particular, a semi-analytic theory of the motion of the Moon around the Earth and the libration of the lunar equatorial plane using different reference frames are introduced. The second part of this article shows that the choice of the canonical Poincaré variables lead to equations in closed form without singularities in e = 0 or I = 0. We introduce new expressions that are sufficiently compact to be used for the study of any artificial satellite. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Approximate analytic method for high-apogee twelve-hour orbits of artificial Earth’s satellites

We propose an approach to the study of the evolution of high-apogee twelve-hour orbits of artificial Earth’s satellites. We describe parameters of the motion model used for the artificial Earth’s satellite such that the principal gravitational perturbations of the Moon and Sun, nonsphericity of the Earth, and perturbations from the light pressure force are approximately taken into account. To solve the system of averaged equations describing the evolution of the orbit parameters of an artificial satellite, we use both numeric and analytic methods. To select initial parameters of the twelve-hour orbit, we assume that the path of the satellite along the surface of the Earth is stable. Results obtained by the analytic method and by the numerical integration of the evolving system are compared. For intervals of several years, we obtain estimates of oscillation periods and amplitudes for orbital elements. To verify the results and estimate the precision of the method, we use the numerical integration of rigorous (not averaged) equations of motion of the artificial satellite: they take into account forces acting on the satellite substantially more completely and precisely. The described method can be applied not only to the investigation of orbit evolutions of artificial satellites of the Earth; it can be applied to the investigation of the orbit evolution for other planets of the Solar system provided that the corresponding research problem will arise in the future and the considered special class of resonance orbits of satellites will be used for that purpose.     

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula’s expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane’s expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long-term motion of the highly elliptical and critically-inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.     

Coordinate systems and lunar observing station positions

Satellite geodesy has yielded the locations of more than fifty stations in a single coordinate system referred to the Earth's center of mass with accuracies in the five to ten meter range. The following different methods have been used at Goddard to accomplish this.Dynamical solutions have been obtained for the locations of some fifty key stations using data from the GEOS satellite program. The distribution of observations about the stations is illustrated in terms of the data obtained for a typical station such as the one at Edinburg, Texas. Geopotential coefficients were held fixed in these solutions. The results of these dynamical determinations implied geodetic datum shifts which were then used to arrive at positions for some two hundred additional stations.Another approach involved the adjustment of the coordinates of seventeen stations on the basis of observations of short arcs of GEOS satellite orbits. These results were found to be consistent with those obtained through ground surveys to about five meters rms in each coordinate.Simultaneous solutions for station locations and geopotential coefficients have also yielded values for positions of some sixty stations, again in a coordinate system defined in terms of the Earth's center of mass.Lunar laser ranging and lunar occultation observing programs involve knowledge of the positions of the observing sites. In some cases the lunar observing program itself yields station coordinate information. In other cases greater reliance is placed upon independent determinations of site locations. The location of an occultation observation site at Olifantsfontein, for example, has been obtained in a center-of-mass system in both the dynamical and simultaneous satellite solutions. It is anticipated that a dynamical satellite solution will be extended in 1973 to obtain center-of-mass coordinates for a station in New Zealand. This will make it possible to tie an occultation site in that region to a dynamically determined coordinate system referred to the mass center. Coordinates for stations at Organ Pass, New Mexico, determined in both the dynamical and simultaneous solutions, and Edinburg, Texas, found in both the dynamical and short-arc adjustments, provide the basis for referring the location of a facility such as the McDonald Observatory to a center-of-mass system either through accurate ground surveying techniques or by means of a satellite geodesty tie. The latter approach has already been used, for example, to fix the position of an isolated site on Madagascar relative to a reference point in Africa and, in turn, to a center-of-mass coordinate system.Estimates of the accuracies of the satellite determinations are discussed.Theoretical aspects of coordinate systems associated with the Earth and the Moon are also considered.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.     

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。     

Gravitational perturbations of equatorial orbits

Asymptotic solutions are developed for the motion of a geocentric satellite in the equatorial plane due to gravitational perturbations such as nonsphericity (especially oblateness) of the primary body. Axisymmetric potentials are considered. A class of transformations is developed and the equations of motion are solved by the method of generalized multiple scales. Further it is shown that the equations of motion can be transformed into the required form to within any specified degree of accuracy. The transformations form an Abelian group of infinite order which leaves the differential equations of motion invariant. Solutions are developed in terms of elementary functions instead of elliptic or other higher transcendental functions and are shown to agree with known results.This investigation was carried out under NASA Grant NGR-31-001-152 with the author as a consultant to Princeton University.     

Averaging the Equations of a Planetary Problem in an Astrocentric Reference Frame

A system of averaged equations of planetary motion around a central star is constructed. An astrocentric coordinate system is used. The two-planet problem is considered, but all constructions are easily generalized to an arbitrary number N of planets. The motion is investigated in modified (complex) Poincarécanonical elements. The averaging is performed by the Hori–Deprit method over the fast mean longitudes to the second order relative to the planetary masses. An expansion of the disturbing function is constructed using the Laplace coefficients. Some terms of the expansion of the disturbing function and the first terms of the expansion of the averaged Hamiltonian are given. The results of this paper can be used to investigate the evolution of orbits with moderate eccentricities and inclinations in various planetary systems.     

The improvement of the lunar ephemeris

At present the fundamental lunar ephemeris is based on Brown's theory of the motion of the Moon with improvements based on the bypassing of Brown's Tables, the removal of the great empirical term, the substitution of the relevant constants of the IAU system of astronomical constants and the retransformation of Brown's series in rectangular coordinates to spherical coordinates. Even so this ephemeris does not represent adequately the recent range and range-rate radio observations, and it will be inadequate for use in the analysis of laser observations of corner reflectors on the Moon. Numerical integrations for these purposes have already been made at the Jet Propulsion Laboratory, but improved theoretical developments are also required; new solutions of the main problem are in hand elsewhere. Work at H.M. Nautical Almanac Office is aimed at obtaining improved values of the constants of the lunar orbit by a rediscussion of occultation observations made since 1943 and at the redevelopment of the series for the planetary perturbations using more precise theories of the motion of the Sun and planets. The techniques and preliminary results of exploratory numerical integrations were briefly described.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, 17–23 August, 1969.     

Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2015

This report continues the practice where the IAU Working Group on Cartographic Coordinates and Rotational Elements revises recommendations regarding those topics for the planets, satellites, minor planets, and comets approximately every 3 years. The Working Group has now become a “functional working group” of the IAU, and its membership is open to anyone interested in participating. We describe the procedure for submitting questions about the recommendations given here or the application of these recommendations for creating a new or updated coordinate system for a given body. Regarding body orientation, the following bodies have been updated: Mercury, based on MESSENGER results; Mars, along with a refined longitude definition; Phobos; Deimos; (1) Ceres; (52) Europa; (243) Ida; (2867) ?teins; Neptune; (134340) Pluto and its satellite Charon; comets 9P/Tempel 1, 19P/Borrelly, 67P/Churyumov–Gerasimenko, and 103P/Hartley 2, noting that such information is valid only between specific epochs. The special challenges related to mapping 67P/Churyumov–Gerasimenko are also discussed. Approximate expressions for the Earth have been removed in order to avoid confusion, and the low precision series expression for the Moon’s orientation has been removed. The previously online only recommended orientation model for (4) Vesta is repeated with an explanation of how it was updated. Regarding body shape, text has been included to explain the expected uses of such information, and the relevance of the cited uncertainty information. The size of the Sun has been updated, and notation added that the size and the ellipsoidal axes for the Earth and Jupiter have been recommended by an IAU Resolution. The distinction of a reference radius for a body (here, the Moon and Titan) is made between cartographic uses, and for orthoprojection and geophysical uses. The recommended radius for Mercury has been updated based on MESSENGER results. The recommended radius for Titan is returned to its previous value. Size information has been updated for 13 other Saturnian satellites and added for Aegaeon. The sizes of Pluto and Charon have been updated. Size information has been updated for (1) Ceres and given for (16) Psyche and (52) Europa. The size of (25143) Itokawa has been corrected. In addition, the discussion of terminology for the poles (hemispheres) of small bodies has been modified and a discussion on cardinal directions added. Although they continue to be used for planets and their satellites, it is assumed that the planetographic and planetocentric coordinate system definitions do not apply to small bodies. However, planetocentric and planetodetic latitudes and longitudes may be used on such bodies, following the right-hand rule. We repeat our previous recommendations that planning and efforts be made to make controlled cartographic products; newly recommend that common formulations should be used for orientation and size; continue to recommend that a community consensus be developed for the orientation models of Jupiter and Saturn; newly recommend that historical summaries of the coordinate systems for given bodies should be developed, and point out that for planets and satellites planetographic systems have generally been historically preferred over planetocentric systems, and that in cases when planetographic coordinates have been widely used in the past, there is no obvious advantage to switching to the use of planetocentric coordinates. The Working Group also requests community input on the question submitting process, posting of updates to the Working Group website, and on whether recommendations should be made regarding exoplanet coordinate systems.     

Expansion of the Hamiltonian for the planetary problem into a Poisson series in the heliocentric reference frame

An expansion of the Hamiltonian for the N-planet problem into a Poisson series using a system of modified (complex) Poincare´ canonical elements in the heliocentric coordinate system is constructed. The Lagrangian and Hamiltonian formalisms are used. The first terms in the expansions of the principal and complementary parts of the disturbing function are presented. Estimates of the number of terms in the presented expansions have been obtained through numerical experiments. A comparison with the results of other authors is made.     

Effects of motion of the equatorial plane on the orbital elements of an earth satellite

Exact differential equations relating the perturbations to satellite orbital elements by the motion of the Earth's equatorial plane are derived, and they are solved to second order in precession. The system proposed in a previous paper (Kozai, 1960), in which the inclination and the argument of perigee are referred to the equator of date and the longitude of the ascending node is measured from a fixed point along a fixed plane and then along the equator of date, can still be recommended for precise studies of satellite motion even when the second-order perturbations are taken into account.     

The long period behavior of the orbits of the Galilean satellites of Jupiter

Some of the results of an investigation into the long period behavior of the orbits of the Galilean satellites of Jupiter are presented. Special purpose computer programs were used to perform all the algebraic manipulations and series expansions that are necessary to describe the mutual interactions among the satellites.The disturbing function was expanded as a Poisson series in the modified Keplerian elements referred to a Jovicentric coordinate system. The differential equations for the modified Keplerian elements were then formed, and all short period perturbations were removed using Kamel's perturbation method. Approximate analytical solutions for these differential equations are derived, and the general form of the solutions are given.     

Strongly perturbed quasi-periodic dynamical systems

It is almost impossible to construct a general theory of the motion of a strongly perturbed dynamical system using classical perturbation theory because this approach uses a reference orbit (e.g. a Keplerian ellipse) which is very different from the actual orbit.A general method, pioneered by Jefferys, is presented here. This method allows each quasi-periodic orbit (for instance a strongly perturbed two body problem: JVIII is the typical example) to specify the coordinates to be used. These coordinates are discovered by a truncated infinite series of coordinate transformations. The transformations are implemented using the idea that the nature of a dynamical system is embodied in the symplectic form. The method is illustracted by a simple example.With modern algebraic and series manipulation languages on present day computers all one needs to begin using this approach is a good numerical integration, the end product being a series for each coordinate. Further weak perturbations are easily incorporated into this semi-analytical solution by the usual methods.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

The singly averaged elliptical restricted three-body problem

The plane, singly averaged, elliptical restricted three body problem is considered in the article. The first three terms are taken in the perturbing function. The equations of motion in terms of the canonical elements of Delaunay are obtained. And the change of the elements of motion of the satellite due to the perturbing function is calculated. An application is given in the case of a satellite in the earth-moon system.     

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.