Theories of lunar libration

1977 ◽  
Vol 284 (1326) ◽  
pp. 573-585 ◽  
Keyword(s):  
Equations Of Motion ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
Computer Laboratory ◽  
The Earth ◽  
Lunar Laser ◽  
Highly Nonlinear ◽  
Librational Motion ◽  
Measured Distance ◽  
Nonlinear Equations Of Motion

The measured distance between a point on the Moon and an observatory on the Earth varies with the librational motion of the Moon about her centre of mass. The motion is caused by the varying attraction of the Earth, Sun and planets upon the Moon and obeys highly nonlinear equations of motion. Because of the high precision of measurements with lunar laser ranging systems, the theory of the motion must be worked out in great detail and the absence of adequate developments limits the interpretation of lunar ranging observations. Numerical integration of the equations of motion is carried out at the Jet Propulsion Laboratory and Eckhardt has developed a semi-literal theory in which coefficients of periodic terms are calculated numerically. There is still need, however, for a literal theory. A brief account will be given of a new literal theory, the algebraic manipulations for which are being carried out by the Camal machine algebraic program developed in the Computer Laboratory at Cambridge. The third harmonic terms in the gravitational potential of the Moon are included and it is intended to include the effect of the Sun.

Effect of non-tidal station loading on Lunar Laser Ranging observatories and on the estimation of Earth orientation parameters

2021 ◽  
Author(s):  
Vishwa Vijay Singh ◽  
Liliane Biskupek ◽  
Jürgen Müller ◽  
Mingyue Zhang
Keyword(s):  
Research Centre ◽  
Laser Ranging ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
Earth Orientation Parameters ◽  
Tidal Station ◽  
Earth Orientation ◽  
The Earth ◽  
Lunar Laser ◽  
Orientation Parameters

<p>The distance between the observatories on Earth and the retro-reflectors on the Moon has been regularly observed by the Lunar Laser Ranging (LLR) experiment since 1970. In the recent years, observations with bigger telescopes (APOLLO) and at infra-red wavelength (OCA) are carried out, resulting in a better distribution of precise LLR data over the lunar orbit and the observed retro-reflectors on the Moon, and a higher number of LLR observations in total. Providing the longest time series of any space geodetic technique for studying the Earth-Moon dynamics, LLR can also support the estimation of Earth orientation parameters (EOP), like UT1. The increased number of highly accurate LLR observations enables a more accurate estimation of the EOP. In this study, we add the effect of non-tidal station loading (NTSL) in the analysis of the LLR data, and determine post-fit residuals and EOP. The non-tidal loading datasets provided by the German Research Centre for Geosciences (GFZ), the International Mass Loading Service (IMLS), and the EOST loading service of University of Strasbourg in France are included as corrections to the coordinates of the LLR observatories, in addition to the standard corrections suggested by the International Earth Rotation and Reference Systems Service (IERS) 2010 conventions. The Earth surface deforms up to the centimetre level due to the effect of NTSL. By considering this effect in the Institute of Geodesy (IfE) LLR model (called ‘LUNAR’), we obtain a change in the uncertainties of the estimated station coordinates resulting in an up to 1% improvement, an improvement in the post-fit LLR residuals of up to 9%, and a decrease in the power of the annual signal in the LLR post-fit residuals of up to 57%. In a second part of the study, we investigate whether the modelling of NTSL leads to an improvement in the determination of EOP from LLR data. Recent results will be presented.</p>

scholarly journals MILLIMETRE MOON MEASUREMENT: 50 YEARS OF LUNAR LASER RANGING SINCE APOLLO 11!

2020 ◽  
Vol XLIII-B3-2020 ◽  
pp. 1105-1110
Author(s):  
J. F. Brock
Keyword(s):  
Laser Ranging ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
The Earth ◽  
Space Race ◽  
Lunar Laser ◽  
Space Star ◽  
Gravitational Influence ◽  
Tv Shows ◽  
Set Up

Abstract. Since the dawn of time the Moon has held fascination for the earliest humans who saw it as a natural navigational beacon, a heavenly body to be revered and a poetic inspiration. Ancient art features the Moon as a prominent subject from all epochs and genres. The name “lunatic” infers that it drives men insane. Giant tides and rapid recessions of water are all attributed to its gravitational influence. As a young boy I was thrilled by stories of Moon travel like Jules Verne’s “From the Earth to the Moon” plus TV shows and movies such as “Lost in Space”, “Star Trek” and “Dr. Who.”The Russian-American “Space Race” focussed on the exciting possibility of man landing on the Moon. I cannot forget the live telecast of the Apollo 11 astronauts on the Moon’s surface in 1969 when I was 13 years old. Four years later I decided to be a land boundary surveyor trained in precise measurement for land title creation. My curiosity was alerted to the Apollo 11 laser ranging aspect of the project when the US team set up a bank of retro-reflectors for measurements from powerful devices on the Earth in the same way we Earthly surveyors make our daily measurements using such EDM equipment.In this paper I will describe the techniques and equipment utilised during this accurate Moon positioning project. You will also see the Earth observatories still measuring to five sites on the Moon and some ancient admirable attempts to determine this distance.

scholarly journals LUNAR LASER RANGING TESTS OF THE EQUIVALENCE PRINCIPLE WITH THE EARTH AND MOON

2009 ◽  
Vol 18 (07) ◽  
pp. 1129-1175 ◽  
Author(s):  
JAMES G. WILLIAMS ◽  
SLAVA G. TURYSHEV ◽  
DALE H. BOGGS
Keyword(s):  
Equivalence Principle ◽  
Primary Objective ◽  
Laser Ranging ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
The Earth ◽  
Lunar Laser ◽  
Wide Range ◽  
Near Term ◽  
Ppn Parameters

A primary objective of the lunar laser ranging (LLR) experiment is to provide precise observations of the lunar orbit that contribute to a wide range of science investigations. In particular, time series of the highly accurate measurements of the distance between the Earth and the Moon provide unique information used to determine whether, in accordance with the equivalence principle (EP), these two celestial bodies are falling toward the Sun at the same rate, despite their different masses, compositions, and gravitational self-energies. Thirty-five years since their initiation, analyses of precision laser ranges to the Moon continue to provide increasingly stringent limits on any violation of the EP. Current LLR solutions give (-1.0 ± 1.4) × 10-13 for any possible inequality in the ratios of the gravitational and inertial masses for the Earth and Moon, Δ(MG/MI). This result, in combination with laboratory experiments on the weak equivalence principle, yields a strong equivalence principle (SEP) test of Δ(MG/MI) SEP = (-2.0 ± 2.0) × 10-13. Such an accurate result allows other tests of gravitational theories. The result of the SEP test translates into a value for the corresponding SEP violation parameter η of (4.4 ± 4.5) × 10-4, where η = 4β - γ - 3 and both γ and β are parametrized post-Newtonian (PPN) parameters. Using the recent result for the parameter γ derived from the radiometric tracking data from the Cassini mission, the PPN parameter β (quantifying the nonlinearity of gravitational superposition) is determined to be β - 1 = (1.2 ± 1.1) × 10-4. We also present the history of the LLR effort and describe the technique that is being used. Focusing on the tests of the EP, we discuss the existing data, and characterize the modeling and data analysis techniques. The robustness of the LLR solutions is demonstrated with several different approaches that are presented in the text. We emphasize that near-term improvements in the LLR accuracy will further advance the research on relativistic gravity in the solar system and, most notably, will continue to provide highly accurate tests of the EP.

Equivalence principle for massive bodies in a generalized theory of gravitation. II

1982 ◽  
Vol 60 (11) ◽  
pp. 1556-1560 ◽  
Author(s):  
J. C. McDow ◽  
J. W. Moffat
Keyword(s):  
Equivalence Principle ◽  
Equations Of Motion ◽  
Upper Bounds ◽  
Laser Ranging ◽  
Coordinate Frame ◽  
Lunar Laser Ranging ◽  
The Earth ◽  
Lunar Laser ◽  
Theory Of Gravitation ◽  
Three Body

The three-body equations of motion are examined for periodic perturbations in an earth centered coordinate frame. The lunar-laser-ranging data is then used to place upper bounds on the new l parameters associated with the earth and sun.

An introductory review of ephemerides for lunar laser ranging

1977 ◽  
Vol 284 (1326) ◽  
pp. 461-466
Keyword(s):  
Numerical Integration ◽  
Laser Ranging ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
Fields Of Study ◽  
The Earth ◽  
Lunar Laser ◽  
Laser Systems ◽  
Orbital Data ◽  
Introductory Review

Precise predictions of the ranges of the retroreflectors on the Moon from the observing stations on the Earth are required to facilitate the making of observations and also to provide a sound basis for the analysis of the observations. The precision of observations is already such that the theories of the Moon’s motion and libration currently used for the ephemerides in the Astronomical Ephemeris are inadequate for the analysis, and so the orbital data are generated by numerical integration. New laser systems will give a further improvement in precision, and further factors will have to be taken into account in the predictions. The exploitation of the data will require the development of new analytical theories, but the results will be of value in many different fields of study.

scholarly journals Dynamical Reference Frames in the Planetary and Earth-Moon Systems

1990 ◽  
Vol 141 ◽  
pp. 173-182
Author(s):  
E. M. Standish ◽  
J. G. Williams
Keyword(s):  
Reference Frame ◽  
Reference Frames ◽  
Optical Data ◽  
Laser Ranging ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
The Earth ◽  
Lunar Laser ◽  
Mariner 9 ◽  
Radar Ranging

We summarize our previous estimates of the accuracies of the ephemerides. Such accuracies determine how well one can establish the dynamical reference frame of the ephemerides. Ranging observations are the dominant data for the inner four planets and the Moon: radar-ranging for Mercury and Venus; Mariner 9 and Viking spacecraft-ranging for the Earth and Mars; lunar laser-ranging for the Moon. Optical data are significant for only the five outermost planets. Inertial mean motions for the Earth and Mars are determined to the level of 0.″003/cty during the time of the Viking mission; for Mars, this will deteriorate to 0.″01/cty or more after a decade or so; similarly, the inclination of the martian orbit upon the ecliptic was determined by Viking to the level of 0.″001. Corresponding uncertainties for Mercury and Venus are nearly two orders of magnitude larger. For the lunar mean motion with respect to inertial space, the present uncertainty is about 0.″04/cty; at times away from the present, the uncertainty of 1′/cty2 in the acceleration of longitude dominates. The mutual orientations of the equator, ecliptic and lunar orbit are known to 0.″002. The inner four planets and the Moon can now be aligned with respect to the dynamical equinox at a level of about 0.″005.

McDonald observatory lunar laser ranging: Beginning the second 25 years

1996 ◽  
Vol 172 ◽  
pp. 409-414
Author(s):  
P.J. Shelus ◽  
R.L. Ricklefs ◽  
J.G. Ries ◽  
A.L. Whipple ◽  
J.R. Wiant
Keyword(s):  
Laser Pulse ◽  
Travel Time ◽  
Lunar Surface ◽  
Laser Ranging ◽  
The Moon ◽  
Round Trip ◽  
Lunar Laser Ranging ◽  
The Earth ◽  
Lunar Laser ◽  
Rich Spectrum

Lunar laser ranging (LLR) (Dickey et al., 1994) consists of measuring changes in the round-trip travel time for a laser pulse traveling between a transmitter on the Earth and a reflector on the Moon. The lunar surface reflectors are still operating normally after almost three decades of use. The ranging data exhibit a rich spectrum of change due to many effects.

scholarly journals WILL IT BE POSSIBLE TO MEASURE INTRINSIC GRAVITOMAGNETISM WITH LUNAR LASER RANGING?

2009 ◽  
Vol 18 (08) ◽  
pp. 1319-1326 ◽  
Author(s):  
LORENZO IORIO
Keyword(s):  
Orbital Motion ◽  
Radial Component ◽  
Laser Ranging ◽  
The Moon ◽  
Lunar Laser Ranging ◽  
The Earth ◽  
Gravitomagnetic Field ◽  
Lunar Laser ◽  
Short Period ◽  
Orbital Revolution

In this paper we mainly explore the possibility of measuring the action of the intrinsic gravitomagnetic field of the rotating Earth on the orbital motion of the Moon with the lunar laser ranging (LLR) technique. Expected improvements in it should push the precision in measuring the Earth–Moon range to the mm level; the present-day root mean square (RMS) accuracy in reconstructing the radial component of the lunar orbit is about 2 cm; its harmonic terms can be determined at the mm level. The current uncertainty in measuring the lunar precession rates is about 10-1 milliarcseconds per year. The Lense–Thirring secular — i.e. averaged over one orbital period — precessions of the node and the perigee of the Moon induced by the Earth's spin angular momentum amount to 10-3 milliarcseconds per year, yielding transverse and normal shifts of 10-1-10-2 cm yr-1. In the radial direction there is only a short-period — i.e. nonaveraged over one orbital revolution — oscillation with an amplitude of 10-5 m. Major limitations come also from some systematic errors induced by orbital perturbations of classical origin, such as the secular precessions induced by the Sun and the oblateness of the Moon, whose mismodeled parts are several times larger than the Lense–Thirring signal. The present analysis holds also for the Lue–Starkman perigee precession due to the multidimensional braneworld model by Dvali, Gabadadze and Porrati (DGP); indeed, it amounts to about 5 × 10-3 milliarcseconds per year.

scholarly journals Que pourra-t-on deduire des mesures de distance terre-lune par laser?

1974 ◽  
Vol 61 ◽  
pp. 269-274
Author(s):  
J. Kovalevsky
Keyword(s):  
Major Axis ◽  
Radio Interferometry ◽  
The Moon ◽  
Semi Major Axis ◽  
Lunar Laser Ranging ◽  
The Earth ◽  
Lunar Laser ◽  
Observational Errors ◽  
External Check

Although several lunar laser ranging stations exist, only one is now fully operational: the McDonald station with internal observational errors of less than 15 cm. The interpretation of the data involves a great number of parameters relative to the Earth and the Moon which are listed.The lunar laser is particularly fit for those parameters that pertain to the Moon, and with future lasers accurate to 2 or 3 cm, it may be expected that this accuracy will be projected into these parameters. The probable determination of the semi-major axis to 1 cm accuracy for a few months mean would imply a new means of determining the non conservative part of the motion of the Moon. A similar precision is to be expected for the rotation of the Moon. The situation for the Earth parameters (Earth rotation and polar motion) is not so good, because of a rather weak geometry of the problem and the monthly one week gap in the observations. Nevertheless, it will give a very useful external check on other competing methods (radio-interferometry, laser or radio-satellites).

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

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