Benefit of New High-Precision LLR Data for the Determination of Relativistic Parameters
AArticle
Benefit of New High-Precision LLR Data for theDetermination of Relativistic Parameters
Liliane Biskupek , Jürgen Müller and Jean-Marie Torre Institute of Geodesy, Leibniz University Hannover, Schneiderberg 50, 30167 Hannover, Germany Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, IRD, Géoazur, Caussols, France * Correspondence: [email protected]
Abstract:
Since 1969, Lunar Laser Ranging (LLR) data have been collected by various observatories andanalysed by different analysis groups. 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 dataover the lunar orbit and the observed retro-reflectors on the Moon. This is a great advantage for variousinvestigations in the LLR analysis. The aim of this study is to evaluate the benefit of the new LLR datafor the determination of relativistic parameters. Here we show current results for relativistic parameterslike a possible temporal variation of the gravitational constant ˙ G / G = ( − ± ) × − yr − , theequivalence principle with ∆ (cid:0) m g / m i (cid:1) EM = ( − ± ) × − and the PPN parameters β − =( ± ) × − and γ − = ( ± ) × − . The results show a significant improvement in theaccuracy of the various parameters, mainly due to better coverage of the lunar orbit, better distribution ofmeasurements over the lunar retro-reflectors, and last but not least, higher accuracy of the data. Withinthe estimated accuracies, no violation of Einstein’s theory is found and the results set improved limits forthe different effects. Keywords: lunar laser ranging, gravitational constant, equivalence principle, PPN parameters
1. Introduction
It was July 20th, 1969 when the astronauts of the Apollo 11 crew landed in the southern part of MareTranquillitatis on the Moon. They deployed the Apollo Lunar Surface Experiments Package, where theretro-reflector for Lunar Laser Ranging (LLR) is now the last operating part of the experiment. Until1973, four further reflectors were deployed on the lunar surface: two reflectors by the astronauts of theApollo 14 and 15 missions, and two reflectors mounted on the unmanned Soviet Lunokhod rovers. Formore than 50 years there is continuous measuring of the distance between observatories on the Earthand retro-reflectors on the Moon. The measurement of round trip travel times between Earth and Moonwith short laser pulses is challenging. The average number of returning photons is less than one per laserpulse [1,2], mainly because of the beam divergence of the laser pulses due to the atmospheric turbulenceand diffraction effects of the reflectors [3]. Further signal loss occurs in the paths of the transmittingand detection optics, in the atmosphere and due to the reflectivity of the retro-reflectors [4]. A series ofsingle measurements over 5-15 minutes is used to calculate a so-called normal point (NP) which is theobservable in the LLR analysis [5]. The observatories on the Earth, that were or are capable to range to theMoon are the Observatoire de la Côte d’Azur (OCA) in France, the McDonald observatory (MLRS) andthe Apache Point Lunar Laser-ranging Operation (APOLLO) in the USA, the Lunar Ranging Experiment(LURE) of the Haleakala observatory on Hawaii, the Matera Laser Ranging Observatory (MLRO) in Italyand the Wettzell Laser Ranging System (WLRS) in Germany. From the end of the eighties OCA startedto investigate the measurement of laser round trip travel time from the observatory to the Moon and
Submitted to
Universe , pages 1 – 13 a r X i v : . [ g r- q c ] D ec of 13 back with laser emitting in the infra-red (IR) at a wavelength λ = β and γ [8,13,14]. By including the new high-precision NP measured with IR wavelengthinto the LLR analysis, improvements for various parameters are expected. The benefit of that NP isinvestigated here in more detail for the parameters related to testing Einstein’s theory.
2. LLR analysis
The analysis model used in LUNAR is based on Einstein’s theory of relativity. It is fully relativistic andcomplete up to the first post-Newtonian (1/ c ) level. To take advantage of the high-precession NP that canbe obtained with an accuracy of several millimetres [2], the LUNAR software was updated continuously[15,16]. Now the LLR analysis model take into account, among other things, gravitational effects of theSun and planets with the Moon as extended body, the higher-order gravitational interaction between Earthand Moon as well as effects of the solid Earth tides on the lunar motion. The basis for the modelled lunarrotation is a 2-layer core-mantle model according to the DE430 ephemeris. A recent overview of LUNARis given in [13], a detailed description can be found in [17].The measured laser travel time τ meas from Earth to Moon and back gives, together with the speed oflight c , the length of the path (forth and back) ρ meas = τ meas · c . (1)Figure 1 shows the principle of LLR measurements in the moving Earth-Moon system. In a weightedleast-squares adjustment the measured length of the light path is compared with the modelled lengthcomputed as ρ model = ρ + ρ + ∆ τ · c (2)where ρ denotes the light path between the observatory at time t and the reflector at time t with ρ = | x EM ( t ) + x ref ( t ) − x obs ( t ) | (3)and the light path ρ between the reflector at time t and the observatory at time t with ρ = | x ME ( t ) − x ref ( t ) + x obs ( t ) | . (4) of 13 ∆ τ of equation (2) takes into account corrections of the light travel time, like the delay due to the lightpropagation through the gravitational potential of Sun and Earth [18,19], an atmospheric delay [20,21], asynodic modulation of the lunar orbit due to radiation pressure [22] and some time- and station-dependentbiases. G G S x yzx yzx yz t t t ρ ρ xyz x E x M B x obs x ref x obs x EM Figure 1.
Scheme of a LLR measurement in the moving Earth-Moon system with the solar system barycenterB, geocenter G and selenocenter S, and the transmitting time t , reflection time t and detection time t . The vectors are x E as barycentric vector to the geocenter at time t , x M as barycentric vector to theselenocenter at time t , x EM = x M − x E as Earth-Moon vector of the outgoing light path, x obs as vectorgeocenter-observatory in the barycentric system at the times t and t , x ref as vector selenocenter-reflector inthe barycentric system at the time t . ρ , ρ as length of outgoing and incoming light path. x , y , z indicatethe different reference systems of the bodies and the solar system [13]. For the calculation of equations (3) and (4) the positions of the observatories and retro-reflectorsare needed in an inertial reference system, here the barycentric celestial reference system (BCRS) withthe barycentric dynamical time (TDB) is selected. Therefore in a first step, effects at the coordinates ofthe observatories and retro-reflectors will be considered according to [23], e.g. tidal effects due to theatmosphere, ocean and solid Earth, and the tectonic movement in the respective body-fixed referencesystem. In a second step, the transformations from the body-fixed systems to the BCRS are carriedout. Here the needed rotation of the Earth is modelled as defined in [15,23], the rotation of the Moon iscomputed simultaneously with the translation corresponding to [24]. The barycentric position and velocityvectors of Earth and Moon are derived from an ephemeris computation of the solar system bodies (allplanets, the Moon and the largest asteroids). Initial values for the computation are taken from the DE421ephemeris [25]. of 13 w r m s [ c m ] Figure 2.
Annual weighted rms (wrms) of the one-way post-fit residuals for 27 485 NP for the time spanApril 1970 to April 2020.
The measured NP serve as observations in the analysis. They are treated as uncorrelated for thestochastic model of the least squares adjustment and are weighted according to their accuracy. Theadjustment is done in a Gauss-Markov model where up to 250 unknown parameters can be determinedwith their uncertainties. As result of the analysis one gets the post-fit residuals, of which the weightedroot-mean-square (wrms) is given in Figure 2 for each year. Beginning in 1970 with a wrms of more than25 cm, improvements in the measurement system lead to a decreasing wrms. Since 2006 it is about 2 cm orless.
3. Distribution of the normal points
The distribution of LLR NP has a big impact on the determination of various parameters. Furthermore,non-uniform data distribution are one reason for correlations between solution parameters [26]. Therefore,the distribution of the LLR data is investigated in more detail below with respect to each of theobservatories, retro-reflectors, and synodic angle. For the current study, NP for the period April 1970 toApril 2020 were used. In a pre-analysis, all were investigated for possible outliers. Outliers are defined asNP whose residuals for the Earth-Moon distance exceed a limit of a few decimetres. Where the limit lies isdetermined differently for each observatory because they observe with different accuracies. SometimesNP of one or more nights are shifted by the same amount, e.g. due to calibration problems during themeasurement. Then a correction value is introduced into the analysis for this period. Furthermore, thestandardized normal distribution is used for the evaluation of the outliers. If this distribution exceeds acertain value, the NP is also classified as an outlier and not included in the further analysis. After thepre-analysis, 17 NP of the current data set were identified as outliers. Thus, 27 485 NP for the period April1970 to April 2020 are included in the investigation, 22 021 measured with green and 5464 with IR laserlight. of 13 N u m b e r o f n o r m a l p o i n t s MLRS (25%)LURE (3%)OCA green (42%)OCA IR (20%)APOLLO (9%)MLRO (<1%)WLRS (<1%)
Figure 3.
Distribution of the 27 485 normal points over the the time span April 1970 - April 2020. In thelegend the percentages of the contribution of the respective observatories are given. The three observatoriesMcDonald, MLRS1 and MLRS2 are linked in the analysis and listed here as MLRS.
Figure 3 shows the temporal distribution of the measured NP over the last 50 years. One can see inthe legend, that more then 60 % of the NP were observed by OCA (42 % with green and 20 % with IR laserlight). In the last years only OCA and APOLLO provided regular NP, some NP also came from MLRO andWLRS. For the year 2019, 91 % of the NP were measured by OCA in IR. It is clear, that, with this distributionof NP, the analysis is dominated by the OCA NP. Looking at the distribution of the NP according to therespective reflectors, Figure 4 shows it for the whole time span in the left pie. Here the clear domination ofApollo 15 is obvious. Because Apollo 15 gives the strongest reflected signal due to its large size, it wasmore often tracked by the observatories in the past. This was not beneficial for data analysis. In recentyears and with IR NP the situation has improved considerably. For 2019 (shown in Figure 4, right pie) allretro-reflectors were measured approximately equally often, because of the advantage of IR laser light [1].That is also a big advantage for the analysis, especially for the determination of the lunar libration.
Figure 4.
Distribution of all NP as measurement to the respective reflectors for the whole data span of LLR(left) and for the year 2019 (right). of 13 % o f N P greenIR new Moon full Moon new Moon Figure 5.
Distrubution of the NP over the synodic month. Given are the percentage of NP of the totalnumber of measurements with the specific laser color. Full and new Moon indicates the phases of theMoon.
With the better performance of the measurements now also ranging near new and full Moon ispossible [1] for OCA and WLRS. This leads to a better coverage of the lunar orbit over the synodic month.The synodic month is the time span, when Sun, Earth, and Moon are in the same constellation again. Toillustrate the better coverage, Figure 5 shows the percentage of the NP measured for a specific angle of thesynodic month. In green the measurements with green laser light and in red measurements with IR laserlight are given. In the past with only green NP, there were gaps in the phases of new and full Moon. Nowthere are many more observations in IR and the advantage of it is obvious. The more equal distribution ofthe NP over the synodic month leads to a better coverage of the lunar orbit and is a big benefit for thedetermination of various parameters.
4. Relativistic parameters
The analysis of LLR observations is based on Einstein’s theory of relativity. Thus, theEinstein-Infeld-Hoffmann equation, the signal propagation in the gravitational field of Earth and Sun,the temporal and spatial reference systems as well as their respective transformations are formulatedrelativistically [17]. By modification of the Einstein-Infeld-Hoffmann equation the relativistic parts areexamined in more detail. Recent results, e.g. on the equivalence principle, Yukawa term, metric parametersand geodetic precession, can be found in publications [8,13,14,27]. [28] used LLR data to test parametersof the standard-model extension (SME). The various relativistic model contributions cause significantperiodic variations, e.g. annual, monthly, linked to the node of the moon and combined periods in theEarth-Moon distance, through which it is possible to distinguish them from each other [29]. Due to thelarge distance between Earth and Moon and the effect of the bodies in the solar system, the relativisticeffects in the measured Earth-Moon distance are larger than, e.g. in distance measurements to satellites(SLR) [30]. This is a great advantage of LLR. Also, the long time span of LLR data (> 50 years) is verybeneficial to determine and de-correlate certain relativistic parameters.To determine relativistic parameters in the LLR analysis a two step strategy is used. In the first stepthe non-relativistic Newtonian parameters of the LUNAR model are calculated in a so called standardsolution. Here the relativistic parameters are fixed to their values of Einstein’s theory. The second stepallow the estimation of individual relativistic parameters together with the Newtonian ones.In the following subsections different relativistic effects are investigated, e.g. the equivalence principle,the temporal variation of the gravitational constant and the PPN parameters γ and β . The aim is to findout to what extent the higher precision IR NP have an impact on the estimation of the related parameters of 13 compared to the results of [13], where a shorter time span (April 1070 - January 2015) with 20 856 NP andmuch less IR NP were used. Table 1 gives an overview with the results of [13] and those of the currentstudy. The basics of estimating the relativistic parameters are already given in [13] and are only brieflydiscussed here. Table 1.
Values for relativistic parameters from two different estimations. In the middle column results of[13], in the right column results of the current estimation.parameter Hofmann and Müller [13] current analysis ∆ m g m i ( − ± ) × − ( − ± ) × − ˙ G / G ( ± ) × − y − ( − ± ) × − y − ¨ G / G ( ± ) × − y − ( ± ) × − y − γ − ( − ± ) × − ( ± ) × − β − ( − ± ) × − ( ± ) × − The equivalence principle (EP) dates back to the 17th century when Galileo Galilei studied theacceleration of two bodies in free fall and found that in the same gravitational field it is independent oftheir shape, mass and composition [31]. The second axiom of Isaac Newton states that the force F resultsfrom the multiplication of an acceleration a and the inertial mass m i as F = m i ∗ a . In the gravitational fieldof the Earth Newton’s law of gravitation is F = m g ∗ g . That leads to the equivalence of the inertial mass m i and the gravitational mass m g . If the equivalence principle is valid, the ratio m i / m g , which is calledthe weak equivalence principle (WEP), is equal to 1. The comparison of the free-fall accelerations of twobodies ( a , a ) leads to the test of the equivalence principle as ∆ aa = ( a − a ) a + a = ( m g / m i ) − ( m g / m i ) [( m g / m i ) + ( m g / m i ) ] /2 ≈ (cid:18) m g m i (cid:19) − (cid:18) m g m i (cid:19) = ∆ m g m i . (5)A violation would lead to a different acceleration of the bodies in the same gravitational field. Toinvestigate the WEP on Earth, sensitive torsion balances and test bodies made of different compositions likeberyllium and titan [32], and rubidium and potassium [33] are used. Recent results from the MICROSCOPEsatellite mission confirmed the WEP at the level of ∆ m g / m i = ( ± ) × − [34]. From the analysis ofLLR data between 1969 and 2017 [8] estimated ∆ m g / m i = ( − ± ) × − .In Einstein’s gravitational theory the WEP is extended to the strong equivalence principle (SEP) dueto the gravitational self-energy U of the bodies. For bodies with astronomical sizes, like Earth and Moon,the SEP can be tested [35] and parametrised with the Nordtvedt parameter η by m g m i = + η UMc (6)with self energy U and mass M for the respective body and the speed of light c . In Einstein’s theoryit holds η =
0. By analysing the LLR data a combined test of the SEP and WEP is possible. Here Earthand Moon are test bodies in the gravitational field of the Sun with gravitational self-energies and differentcomposition. A violation of the EP would cause an additional acceleration of the Moon into the directionof the Sun.For the investigation of a possible violation of the EP with LLR data there are, according to [13], twodifferent way which leads to similar results. Here the focus is on the determination via an additional of 13 acceleration of the Moon ¨ x mgmi with the relative coordinates between Sun and Moon x SM and the distance r SM to the Sun, where the largest perturbation is given by¨ x mgmi = ∆ (cid:18) m g m i (cid:19) EM GM Sun x SM r . (7) GM Sun denotes the gravitational constant times the mass of the Sun. That method keeps the interactionwith all other forces in the calculation of the ephemeris.The result of [13] for the EP test was ∆ (cid:18) m g m i (cid:19) EM = ( − ± ) × − with correlations of up to 60 % with GM EM because of the dependence on the synodic angle D [36].There are also correlations of up to 60 % with the X-components of the reflector coordinates. Compared tothe result of this study with ∆ (cid:18) m g m i (cid:19) EM = ( − ± ) × − the accuracy was improved and the correlations to GM EM and the reflector coordinates decreased to40 %. Here the better coverage of the LLR NP over the synodic angle, shown in Figure 5, is a clear benefitfor the determination of the EP parameter.[14] investigated a possible violation of the EP due to assumed dark matter in the galactic centrewhich would cause an Earth–Moon range oscillation with a sidereal month period. The amplitude forsuch an oscillation, determined from LLR post-fit residuals, was found to be A = ± From Einstein’s general theory of relativity, it follows that the gravitational constant G is a temporallyand spatially invariable quantity [37]. According to the investigations of [38] and [39], however, theexistence of alternative theories is possible, which allow a variation of the gravitational constant. One ofthe best known is the Brans-Dicke theory, a scalar-tensor theory. It is an extension of Einstein’s theory withadditional scalar fields [38]. Recent studies by [40] and [41] confirm the considerations that a variation ofthe gravitational constant in the range from ˙ G / G = − yr − to ˙ G / G = − − yr − might be possible.According to the remarks of [42], there are also theories which admit so-called preferred reference systems.Also in such systems, a time dependence of G would be possible. The recent upper bounds for a non-zerovalue of ˙ G by using LLR data come from the analysis of the ephemeris of the solar system bodies with˙ G / G = × − yr − [43] and ˙ G / G = × − yr − [44]. From the analysis of MESSENGER data,[45] get an upper limit for ˙ G / G < × − yr − .The estimation of a linear and quadratic part of the gravitational constant as a function of time isdone in the analysis of LLR data with G ( t ) = G (cid:18) + ˙ GG ∆ t +
12 ¨ GG ∆ t (cid:19) (8)as part of the ephemeris calculation. In the standard solution ˙ G = ¨ G = ∆ t results from the current calculation time and the beginning of the LLR measurements. The partial of 13 derivatives of ˙ G and ¨ G needed for the adjustment in the Gauss-Markov model are calculated by numericaldifferentiation of the geocentric lunar ephemeris.The results of [13] for the temporal and quadratic variation were estimated as separate parameters intwo calculations as ˙ GG = ( ± ) × − yr − ,¨ GG = ( ± ) × − yr − .For this results, the initial values of the lunar core rotation vector ω c were fixed to their estimatedstandard solution values because of the high correlation of up to 94 % with ˙ G and ¨ G . High correlations ofup to 83 % with some components of the station coordinates were reduced by introducing constraints onthe estimated station coordinates.The determination of ˙ G and ¨ G with more NP including a high number of IR NP from OCA resulted in˙ GG = ( − ± ) × − yr − ,¨ GG = ( ± ) × − yr − for the separate calculation of the two parts of G . In the current calculation the correlations with partsof the core rotation vector ω c were up to 70 % and decreased compared to [13]. But they were high enoughto affect the determination of ˙ G and ¨ G , therefore they were fixed to their estimated standard solutionvalues. The correlations with station coordinates could be reduced to up to 20 % compared to [13] andno constraints were used in the calculation. Also correlations with other parameters of the Earth-Moonsystem significantly decreased and are now at most 40 % with the Z-component of the initial values of thelunar orbit. Here the benefit of the longer data span with accurate NP leads to the improvement and abetter and independent determination of the linear and quadratic part of G .For the determination of the linear and quadratic part of G together the values are˙ GG = ( ± ) × − yr − ,¨ GG = ( ± ) × − yr − .The parameters are correlated to each other with 70 %. The correlations to other parameters were alsohigher than in the separate estimation. This leads to a less accurate determination of ˙ G and ¨ G . Nevertheless,the accuracies are in a similar range as for the separate estimation and the results underline the validity ofEinstein’s theory in the given limits. β and γ In the framework of the parametrized post-Newtonian (PPN) approximation of Einstein’s theory, theparameter β indicates the non-linearity of gravity and γ the size of space-curvature [42]. Both values areequal to 1 in this theory. Recent analysis of MESSENGER data [45] gives a value for β − = ( − ± ) × − . From theanalysis of solar system ephemeris there are values at the level of 7 × − for β and 5 × − for γ [44].[46] gets values of β − = ( − ± ) × − and γ − = ( ± ) × − .From the analysis of LLR data [13] get values for the determination of β and γ via theEinstein-Infeld-Hoffmann equations of motion as β − =( − ± ) × − , γ − =( − ± ) × − .Both values show high correlations of up to 82 % to station coordinates and the Z-component ofthe lunar initial velocity. Further correlations are to additional rotations between the Earth-fixed andspace-fixed reference systems of up to 47 %. For this reasons, the additional rotation was fixed to the valuesof the standard solution and the station coordinates were constraint.In the current analysis the PPN parameters were determined to β − =( ± ) × − , γ − =( ± ) × − .The correlations to the station coordinates now are up to 60 %, to the Z-component of the lunar initialvelocity 40 % and to the additional rotation 30 %. All of them were reduced significantly. To make theresults more comparable with those of [13], the additional rotations were nevertheless fixed to the valuesof the standard solution. The remaining correlations are now up to 40 % with the previously mentionedparameters. Compared to the results of [13] the values of β improved slightly, γ is on a similar level. Thelonger time span and IR NP are not as beneficial for the estimation of PPN parameters as for the otherrelativistic parameters shown, but the correlations decreased and there is still no violation of Einstein’stheory.
5. Summary and Outlook
The aim of this study was the investigation of the benefit of high-precision IR LLR measurementsfor determining relativistic parameters in comparison to the results of [13]. The model of the Earth-Moonsystem remained the same between the two calculations of relativistic parameters. The only major changescome from the longer time span of the LLR data and from many more measurements in the IR. Fromthe previous discussions, it is clear that the IR data provide a major advantage for the LLR analysis. Theaccuracies of the relativistic parameters could be improved due to the better coverage of the lunar orbitand the accuracy of the data itself. Another advantage is the decorrelation of the relativistic parameterswith other parameters of the Earth-Moon system. A summary of the results can be found in Table 1. So far,from the analysis of the LLR data, the assumptions of Einstein’s theory of relativity have been confirmed,now with improved limits.An expanded network of single corner-cube retro-reflectors (CCRs) to be placed on the lunar surfacenear the limbs and poles from the year 2022 on will improve the existing geometry. Such CCRs are alsobeneficial in terms of thermal resilience and increased return signal strength. This will improve the rangingaccuracy by a factor of 10 to 100 [47].With the construction of the new LLR facility at Table Mountain Observatory (JPL’s OpticalCommunication Testbed Laboratory - OCTL) in California, for the first time, it will be possible to conductdifferential LLR with an expected range precision of less than 30 micrometers - a factor of 200 better than the current accuracy [48]. This opens new possibilities for improved analysis of the whole LLR parameterset. The improvements on the technical side and further measurements in IR will make it possible, forexample, to investigate effects related to the deep lunar interior and rotation and to determine relativisticparameters with higher accuracy. Together with improved modelling of the lunar interior and rotation inthe LUNAR software, this will significantly improve many parameters determined from the analysis ofthe LLR data.
Funding:
This research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)under Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers – 390837967.
Acknowledgments:
Current LLR data are collected, archived, and distributed under the auspices of the InternationalLaser Ranging Service (ILRS) [49]. We acknowledge with thanks that more than 50 years of processed LLR data hasbeen obtained under the efforts of the personnel at the Observatoire de la Côte dAzur in France, the LURE Observatoryin Maui, Hawaii, the McDonald Observatory in Texas, the Apache Point Observatory in New Mexico, the MateraLaser Ranging station in Italy and the Wettzell Laser Ranging Station in Germany.
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