Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models

The aim of this work is to combine the model of orbital and rotational motion of the Moon developed for DE430 with up-to-date astronomical, geodynamical, and geo- and selenophysical models. The parameters of the orbit and physical libration are determined in this work from lunar laser ranging (LLR) observations made at different observatories in 1970–2013. Parameters of other models are taken from solutions that were obtained independently from LLR. A new implementation of the DE430 lunar model, including the liquid core equations, was done within the EPM ephemeris. The postfit residuals of LLR observations make evident that the terrestrial models and solutions recommended by the IERS Conventions are compatible with the lunar theory. That includes: EGM2008 gravitational potential with conventional corrections and variations from solid and ocean tides; displacement of stations due to solid and ocean loading tides; and precession-nutation model. Usage of these models in the solution for LLR observations has allowed us to reduce the number of parameters to be fit. The fixed model of tidal variations of the geopotential has resulted in a lesser value of Moon’s extra eccentricity rate, as compared to the original DE430 model with two fit parameters. A mixed model of lunar gravitational potential was used, with some coefficients determined from LLR observations, and other taken from the GL660b solution obtained from the GRAIL spacecraft mission. Solutions obtain accurate positions for the ranging stations and the five retroreflectors. Station motion is derived for sites with long data spans. Dissipation is detected at the lunar fluid core-solid mantle boundary demonstrating that a fluid core is present. Tidal dissipation is strong at both Earth and Moon. Consequently, the lunar semimajor axis is expanding by 38.20 mm/yr, the tidal acceleration in mean longitude is \(-25.90 {{}^{\prime \prime }}/\mathrm{cy}^2\), and the eccentricity is increasing by \(1.48\times 10^{-11}\) each year.     

Compressed planetary and lunar ephemerides

A package of FORTRAN software has been developed which provides planetary and lunar positions, with respect to the solar system barycenter, for all times in the interval 1801–2049; positions agree to 1 milliarcsecond with those generated by Jet Propulsion Laboratory Development Ephemeris 200 (DE200). The system consists of approximately 800 kilobytes of ephemeris files and 40 kilobytes of programs, totalling 5% of the storage required by DE200. After removal of reference orbits, segments of DE200 positions were fitted by finite Chebyshev series of degree 40. The Chebyshev coefficients were rounded to integer multiples of a suitable unit and packed to form the ephemeris files.     

Improvement of the numerical lunar ephemeris with laser ranging data

Analysis of lunar laser ranging data is underway at several institutions. We describe here our efforts at improving the numerical ephemeris of Moon, based on over three years' span of data. Orbit generation and correction procedures are discussed briefly. Comparisons of the new ephemeris with observations and with a widely available ephemeris are illustrated. The standard deviation of the observation residuals is 7 m.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.     

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.     

Estimation of the lunar physical librations

The recent long-term integration of JPL ephemeris DE403/LE403 yielded lunar physical librations covering 6000 years. A Fourier analysis of a 718-year subset of this span produced estimates of the component frequencies of the forced and free librations. A subsequent iterative least-squares estimation procedure provided precise values for phases and for time-varying amplitudes and frequencies. Two free libration modes were found; presence of a third is possible but close to the noise.     

On the precession expressions based on the de ephemeris

Since 1984, the new IAU (1976) System of Astronomical Constants has become effective; meanwhile, the new lunar and planetary ephemerides (DE200/LE200) have been introduced into the Astronomical Almanac. In order to obtain the best fit of these ephemerides to the observational data, some modifications to the constants were made (Kaplan 1961). The modified values of these constants have been accepted by many users (particularly in the Merit Project), (Melbourne et al. 1983), although there has not been any new resolution of IAU. To avoid these inconsistencies, it seems to be necessary to rediscuss the adopted value of some astronomical constants in the new system. This paper discusses the problems for selection of the precession quantities and derives the precession expressions based on the motion of ecliptic from the DE ephemeris.     

The Moon’s physical librations and determination of their free modes

The Lunar Laser Ranging experiment has been active since 1969 when Apollo astronauts placed the first retroreflector on the Moon. The data accuracy of a few centimeters over recent decades, joined to a new numerically integrated ephemeris, DE421, encourages a new analysis of the lunar physical librations of that ephemeris, and especially the detection of three modes of free physical librations (longitude, latitude, and wobble modes). This analysis was performed by iterating a frequency analysis and linear least-squares fit of the wide spectrum of DE421 lunar physical librations. From this analysis we identified and estimated about 130–140 terms in the angular series of latitude librations and polar coordinates, and 89 terms in the longitude angle. In this determination, we found the non-negligible amplitudes of the three modes of free physical libration. The determined amplitudes reach 1.296′′ in longitude (after correction of two close forcing terms), 0.032′′ in latitude and 8.183′′ × 3.306′′ for the wobble, with the respective periods of 1056.13 days, 8822.88 days (referred to the moving node), and 27257.27 days. The presence of such terms despite damping suggests the existence of some source of stimulation acting in geologically recent times.     

Lunar baselines and libration from differential Vlbi observations of alseps

Differential very-long-baseline interferometric observations of signals from Apollo Lunar Surface Experiment Package telemetry transmitters will yield the relative projected positions of the transmitters with uncertainty of only 1-3 m, set mainly by uncertainty of the lunar ephemeris. Noise and systematic instrumental errors which in the past affected similar observations have been reduced to the equivalent of a few centimeters on the lunar surface by the development of a new type of differential receiver. Continued observations should yield a determination of the motion of the Moon about its center of mass with uncertainty less than 1 s of selenocentric arc. Improvements (by other means) in our knowledge of the Moon's orbital motion would allow a further order-of-magnitude refinement in the libration and relative position results obtainable by differential VLBI.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.     

Detecting the errors in solar system ephemeris by pulsar timing

Pulsar timing uses planetary ephemerides to convert the measured pulse arrival time at an observatory to the arrival time at the Solar System barycenter(SSB). Since these planetary ephemerides cannot be perfect, a method of detecting the associated errors based on a pulsar timing array is developed. By using observations made by an array of 18 millisecond pulsars from the Parkes Pulsar Timing Array, we estimated the vector uncertainty from the Earth to the SSB of JPL DE421, which reflects the offset of the ephemeris origin with respect to the ideal SSB, in different piecewise intervals of pulsar timing data, and found consistent results. To investigate the stability and reliability of our method, we divided all the pulsars into two groups. Both groups yield largely consistent results, and the uncertainty of the Earth-SSB vector is several hundred meters, which is consistent with the accuracy of JPL DE421. As an improvement in the observational accuracy, pulsar timing will be helpful to improve the solar system ephemeris in the future.     

On the orientation of ephemeris reference frames

One may construct complete planetary and lunar ephemerides, referred to the equator and dynamical equinox of some epoch, strictly from ranging data alone. Such an ephemeris would be completely independent from any optical data and therefore independent of any stellar catalogue. By using such an ephemeris to then analyse optical observations, one could theoretically derive many of the pertinent features of the catalogue system to which the optical observations are referred. Such features include the equinox offset, equinox motion and systematic proper motion errors. In practice, the optical observations are used in the fitting process, but essentially the same determinations may be made.This paper presents estimates of the equinox offset and equinox motion of the FK4 as determined by the ephemeris fitting process and compares them with corresponding determinations by Fricke. No significant differences are found. Further, it is indicated how one may also estimate a value for precession and the value of the obliquity from the ephemerides. These' values are also compared with the presently adopted ones.     

Corrections to the improved lunar ephemeris

Based primarily upon the formation of new conditional equations using analytical partial derivatives of the moon's mean elements, meridian circle observations of the moon from 1952–67 have been examined to determine corrections to the constants of lunar theory and to the fundamental coordinate system (FK4). With certain exceptions, the new corrections are in agreement with those published earlier by the author. Systematic corrections to FK4 are surprisingly large, although in agreement with some other recent determinations. New corrections to the lunar ephemeris, resulting from the discussion, are also presented.     

On the accuracy of lunar ephemerides using the data provided by the future Russian lunar laser ranging system

The potential effect of the future Russian lunar laser ranging system (LLRS) on the accuracy of lunar ephemerides is discussed. In addition to the LLRS in Altai, several other observatories suitable for the LLRS installation are considered. The variation of accuracy of lunar ephemerides in the process of commissioning of new LLRS stations is estimated by mathematical modeling. It is demonstrated that the error in the determination of certain lunar ephemeris parameters may be reduced by up to 16% after seven years of operation of the Altai LLRS with a nearly optimal observational program.     

The effect of various solar ephemerides on equator and equinox solutions from observations of the Sun with the reversible transit circle at Cape observatory from 1907 to 1959

Observations of the Sun were made with the Cape reversible transit circle from 1907 to 1959. We have made least squares solutions for six unknowns viz., equator and equinox corrections and corrections to earth orbital parameters including the ephemeris mean longitude of the Sun, the mean obliquity of the ecliptic, the mean longitude of perihelion, and the mean eccentricity of the earth's orbit based on Newcomb's, DE102, and DE200 Ephemerides for each of six catalogs of observations made during that period. The six unknowns are also determined simultaneously for the six catalogs taken together. The six catalogs are absolute, in that methods of observation and reduction were adopted in such a way as to produce a system of results not closely dependent on the adopted system of assumed clock and azimuth star positions.The observed equator and equinox corrections from a comparison of DE200 with the Cape Sun observations referred to an improved FK4 system are –0.07±0.01 arcsec and –0.20±0.04 arcsec, respectively, at the mean epoch of observation, 1933.02. The time rate of change of the equator correction was not significant. The time rate of change of the observed equinox is –1.02±0.30 arcsec per century.The observed equinox correction of the DE102 at 1933.02 is –0.41±0.04 arcsec, which is 0.5 arcsec less than the NEWCOMB (Herget) equinox correction. This confirms the result based on Washington Sun observations.     

Modern Numerical Ephemerides of Planets and the Importance of Ranging Observations for Their Creation

The JPL planetary and lunar ephemerides – DE200/LE200, DE403/LE403, DE405/LE405 and the planetary and lunar ephemerides, EPM87, EPM98, and EPM2000, constructed in the Institute of Applied Astronomy of RAS are described. Common properties and differences of the various ephemerides are given. Graphical comparisons of the DE ephemerides with each other and with the EPM ephemerides are presented. A fairly good agreement of planetary orbits is between DE403, DE405 and EPM98, EPM2000, respectively, over the interval of 120 years (1886–2006) covered by EPM98 and EPM2000. Some differences are explained by a slight disagreement in representing the orbits of Ceres, Pallas, and Vesta as they affect the planets. The accurate radar observations of planets and spacecraft make it possible not only to improve the orbital elements of planets but to determine a broad set of astronomical constants as well: km/AU, parameters of Mars rotation including its precessional rate, the masses of Jupiter, Ceres, Pallas, and Vesta, relativistic parameters of the PPN formalism, the variability of the gravitational constant G. These have been obtained in the fitting process of the DE405 and EPM2000 ephemerides to observational data, including nearly 80000 American and Russian radar observations of planets (1961–1997), ranging and doppler to the Viking and Pathfinder landers, and other miscellaneous measurements from various sources and spacecraft.     

ALSEP-quasar differential VLBI

A program of ALSEP-Quasar Very Long Baseline Interferometry (VLBI) is being carried out at the Jet Propulsion Laboratory. These observations primarily employ a 4-antenna technique whereby simultaneous observations with two antennas at each end of an intercontinental baseline are used to derive the differential interferometric phase between a compact extragalactic radio source (usually a quasar) and a number of ALSEP transmitters on the lunar surface. A continuous ALSEP-quasar differential phase history over a few hour period will lead to milliarcsecond angular accuracy in measuring the lunar position against the quasar reference frame if suitable calibration measurements are obtained. Development of this application of the 4-antenna technique has been underway at JPL for more than a year and is now producing high quality data utilizing Deep Space Network (DSN) stations in Australia, Spain, and Goldstone, California as well as the STDN Apollo station at Goldstone. These high accuracy observations are of value to tie the lunar ephemeris to a nearly inertial extragalactic reference frame, to test gravitational theories, and to measure the Earth-Moon tidal friction interaction.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

Precision of the ephemerides of outer planetary satellites

Ephemerides of planetary satellites are needed to address many problems. These ephemerides are used for subsequent observations. A comparison of the available ephemerides with new observations allows the accuracy of the former to be assessed. However, the precision of the ephemerides must be known a priori when solving the tasks. In this paper we formulate and solve the problem of estimating the precision of the ephemerides of outer planetary satellites derived from observations when applied up to the future moments.The methods of assessing the precision of ephemerides involve producing a set of samples of the same ephemeris inferred from observations with different samples of Monte Carlo generated random errors (RO) superimposed onto it. The statistical parameters of simulated observational errors are based on the results of the reduction of real satellite observations. We compute the deviations of the samples of the ephemeris from the standard ephemeris inferred from real observations and adopt the root-mean-square deviation of the apparent coordinates as the precision of the ephemeris. We also use alternative methods: one based on the matrix of covariances of parameter errors (RP), and another one based on bootstrap samples of observations (BS).We use three methods (RO, RP, and BS) to estimate the precision of the ephemerides of all the 107 outer planetary satellites over the 2010-2020 time interval. The precision of the ephemerides of different satellites varies from 0.05 to 4.0 arcsec. For a number of satellites new observations are of vital importance for maintaining the precision of the ephemerides at a level that would allow identification of satellites during the reduction of observations. For some satellites the precision of their ephemerides is of the order of the sizes of their orbits and such satellites can be considered to have been lost. We show that the method of bootstrap samples (BS) can give doubtful results in the cases where there are few observations, which covered a time interval that is shorter than the orbital period of the satellite.Our results suggest obtaining more precise ephemeris making new observations at the times of maximum estimated errors of the ephemeris.All the inferred estimates of the precision of ephemerides are available from the MULTI-SAT ephemeris server: www.imcce.fr/sat (IMCCE), www.sai.msu.ru/neb/nss/index.htm (SAI).     

A new dynamical model for the Uranian satellites

We have developed a new dynamical model of the main Uranian satellites, based on numerical integration and fitted to astrometric observations. Old observations, as well as modern and Voyager observations have been included. This model has provided ephemerides that have already been used for predicting the mutual events during the PHE-URA campaign. It is updated here to improve the prediction of these events. We also tried to assess the real accuracy of our ephemerides by checking the distance differences of the Uranian satellites, using simultaneously our former and new model. It appears that both solutions are very close to each other (within few tens of kilometers), and most probably accurate at the level of few hundred of kilometers. Using new available meridian observations of the Uranian satellites, we have checked the Uranian ephemeris accuracy using DE406. An error of more than 0.1 arcsec on the Uranian position is observed.     

Accurate analytical representation of Pluto modern ephemeris

An accurate development of the latest JPL’s numerical ephemeris of Pluto, DE421, to compact analytical series is done. Rectangular barycentric ICRF coordinates of Pluto from DE421 are approximated by compact Fourier series with a maximum error of 1.3 km over 1900–2050 (the entire time interval covered by the ephemeris). To calculate Pluto positions relative to the Sun, a development of rectangular heliocentric ICRF coordinates of the Solar System barycenter to Poisson series is additionally made. As a result, DE421 Pluto heliocentric positions by the new analytical series are represented to an accuracy of better than 5 km over 1900–2050.     

Approximation methods in celestial mechanics. Application to Pluto's motion

Various methods of approximation for the computation of planetary perturbations are investigated: Fourier, Fourier-Chebyschev and Legendre. Application is made to Pluto's motion. Based upon JPL's ephemeris DE200, Pluto's mean elements are provided and also a compact ephemeris covering 2 centuries.     

Multiscale Analysis of the Instantaneous Eccentricity Oscillations of the Planets of the Solar System from 13 000 BC to 17 000 AD

Astronomy Letters - The high-resolution Jet Propulsion Laboratory DE431 and DE432 planetary ephemeris are used to evaluate the instantaneous eccentricity functions of the orbits of the planets of...