Abstract
The distance between the observatories on the Earth and the retro-reflectors on the Moon has been regularly measured with Lunar Laser Ranging (LLR) since 1970. In recent years, LLR observations have been carried out at infrared wavelength (OCA, WLRS), resulting in a better distribution of LLR normal points over the lunar orbit and retro-reflectors with a higher accuracy, also leading to a higher number of LLR observations in total. By analysing LLR data, Earth Orientation Parameters (EOPs) can be determined along with other parameters of the Earth-Moon system. Focusing on \(\Delta \)UT1 and terrestrial pole coordinates the accuracies have improved significantly compared to the previous results. In the past, the reported uncertainties of the estimated parameters were published as three times the formal error from the least-squares adjustment to account for small random and systematic errors in the LLR analysis. To investigate if such a scaling factor is still needed, a sensitivity analysis was performed. The current best accuracies are 12.36 \(\upmu \)s for \(\Delta \)UT1, 0.47 mas for \(x_p\) and 0.59 mas for \(y_p\). Also the determined corrections to the long-periodic nutation coefficients of the MHB2000 model are now significantly smaller with higher accuracies, i.e., accuracies better than 0.18 mas are obtained.
You have full access to this open access chapter, Download conference paper PDF
Keywords
1 Introduction
Lunar Laser Ranging (LLR) has been measuring the distance between the Earth and the Moon with laser pulses for more than 53 years. Currently four observatories perform regular measurements: The Côte d’Azur Observatory, France (OCA), the Apache Point Observatory Lunar Laser ranging Operation, USA (APOLLO), the Matera Laser Ranging Observatory, Italy (MLRO) and the Geodetic Observatory Wettzell, Germany (WLRS). In the past also the McDonald Laser Ranging Station, USA (MLRS) and the Lure Observatory on Maui/Hawaii, USA (LURE) contributed to the measurements. On the Moon there are five retro-reflectors where laser pulses from the observatories are reflected back to Earth. The measurement of round trip travel times with short laser pulses over 5 min to 15 min is used to calculate a so-called normal point (NP) (Michelsen 2010) which is the observable in the LLR analysis. With the analysis of the LLR data, contributions to terrestrial, lunar and celestial reference frames (Müller et al. 2009a; Hofmann et al. 2018; Pavlov 2019) as well as the understanding of the lunar interior (Williams et al. 2013; Pavlov et al. 2016) are possible. One major task of LLR is to test the validity of General Relativity in the solar system. Test quantities include, e.g., the equivalence principle, temporal variation of the gravitational constant G, Yukawa term, metric parameters, and geodetic precession (Williams et al. 2012; Viswanathan et al. 2018; Hofmann and Müller 2018; Zhang et al. 2020; Biskupek 2021). Furthermore, the determination of Earth Orientation Parameters (EOPs) is also possible from LLR data. It includes parameters like terrestrial pole coordinates and the Earth rotation phase \(\Delta \)UT1 (Singh et al. 2022a; Biskupek et al. 2022), the celestial pole coordinates (Zerhouni and Capitaine 2009; Cheng et al. 2019) and coefficients for precession and nutation (Hofmann et al. 2018; Biskupek et al. 2012). As special case the Universal Time at a specific location \(\Delta \)UT0 can be determined. \(\Delta \)UT0 and the coefficients of the nutation series are of particular interest, since these parameters are otherwise only determined from Very Long Baseline Interferometry (VLBI).
2 Data and Analysis
The current LLR dataset includes 30172 NPs over the time span April 1970 to April 2022. Starting 2015, many NPs have been measured with laser pulses at infra-red (IR) wavelength, enabling distance measurements near new and full Moon for OCA and WLRS (Chabé et al. 2020; Eckl et al. 2019). This leads to a better coverage of the lunar orbit over the synodic month, i.e. the time span in which Sun, Earth, and Moon return to a similar constellation again. With a better coverage of the lunar orbit, it is possible to estimate various parameters of the Earth-Moon system with higher accuracy and reduced internal correlation. This benefit, together with a higher number of NPs per night, gives the motivation for the determination of EOPs from LLR.
The parameter estimation with the LUNAR analysis software consists of several parts. One part is the calculation of the ephemeris of the eight planets, Sun, Moon, Pluto and asteroids (Ceres, Vesta, and Pallas) as well as the orientation of the core and mantle of the Moon. The needed initial positions and velocities are taken from the DE440 ephemeris (Park 2021). The calculation of the Moon’s rotation is carried out simultaneously with the ephemeris calculation. Another part is the calculation of the Earth-Moon distance. The rotation of the Earth is described by two series of EOPs. For the time span 04.1970 to 01.01.1983 the Kalman Earth Orientation Filter (KEOF) COMB2019 series (Ratcliff and Gross 2020) is used and from 02.01.1983 on the IERS EOP C04 series (Bizouard et al. 2019). The difference between these series is the input data, only the COMB series includes LLR data. For this reason, the series fits the LLR analysis better in the initial phase of the observations. From the 1980s on, the differences between the series are small (only a few mas and ms) and the IERS series is used for its shorter latency. All other models in the LLR analysis follow the recommendations of the IERS Conventions 2010 (Petit and Luzum 2010). The last part of the analysis is the parameter estimation itself with the calculation of the residuals between the observed NPs and calculated Earth-Moon distance in a least-squares adjustment (LSA). The NPs are treated as uncorrelated for the stochastic model of the LSA and are weighted according to their accuracies.
3 Determination of Earth Orientation Parameters
The terrestrial pole coordinates, \(x_p\) and \(y_p\), describe the change of the rotation axis with respect to the Earth’s surface. The Earth rotation phase \(\Delta \)UT1 and the Length of Day (LOD) refer to the rotation of the Earth about its axis. All these parameters are summarised as Earth Rotation Parameters (ERPs). Together with the celestial pole offsets, as corrections to the conventional precession–nutation model, they define the EOPs.
For the analysis of LLR data, the Barycentric Celestial Reference System (BCRS) is used as the inertial system. The coordinates of the observatories and retro-reflectors are given in their respective body-fixed reference systems, like the International Terrestrial Reference System (ITRS) for the Earth and the Principle Axis System (PAS) for the Moon, and are transformed during the analysis into the inertial system. For the Earth, the transformation from ITRS to the Geocentric Celestial Reference System (GCRS) is given by
Here \(\mathbf {W}(dt)\) includes the terrestrial pole coordinates \(x_p\) and \(y_p\). The Earth rotation phase \(\Delta \)UT1 is part of \(\mathbf {R}(dt)\). Finally \(\mathbf {Q}(dt)\), represented here according to the Fukushima–Williams parametrisation via precession and nutation (Fukushima 2003; Williams 1994), contains the coefficients of the nutation series. As the rotation matrix (Eq. 1) is included in the LLR analysis model, the various parameters of the formula can be estimated directly in the least-squares adjustment of the LLR data together with the other parameters of the Earth-Moon system. A more detailed description of the EOP determination from LLR data is given in Biskupek (2015), Hofmann et al. (2018), Singh et al. (2022a), Biskupek et al. (2022).
Another approach to determine ERPs from LLR data is given by Dickey et al. (1985), Müller (1991) and Pavlov (2019), where the ERPs are determined from the post-fit residuals of the least-squares adjustment of LLR data. In this way the variation of longitude \(\Delta \)UT0 can be determined (Chapront-Touzé et al. 2000) by
as combination of \(\Delta \)UT1 and the terrestrial pole coordinates \(x_p, y_p\), with the observatories longitude \(\lambda \) and latitude \(\phi \). The variation of latitude VOL is given by
The disadvantage of this approach is that the correlations between the ERPs and the other parameters of the Earth-Moon system can not be investigated compared to the approach via Eq. (1). However, to better assess the results of the two approaches, they will be compared in a future study.
3.1 Earth Rotation Parameters
In the LLR analysis, different cases for the ERP estimation can be set up, e.g., by selecting certain time spans of data, specific nights on which a minimum number of NPs is available, or selecting NPs from specific observatories. Previous investigations (Singh et al. 2022a; Biskupek et al. 2022) show that the accuracy of the determined ERPs has greatly improved from 2000. Nevertheless, the reported uncertainties of the estimated parameters from the LLR analysis were normally published as three times the formal error from the LSA (\(3\sigma \)). In the past, it was assumed that some small random and systematic errors remained in the LLR modelling and analysis, and affect the determined parameters. To give more realistic uncertainties for the determined parameters and to also consider possible shortcomings in the analysis a scaling factor for the formal errors of the least-squares adjustment was used (Müller 1991; Biskupek 2015; Hofmann et al. 2018; Singh et al. 2022b). Systematic errors include, e.g., the uneven distribution of NPs during the synodic month and the constellation of Earth and Moon when observing an LLR NP, because of the inaccuracy of atmospheric delay models for low altitude observations. Further error sources are the imperfection of lunar ephemeris and rotation, e.g., because of simplified modelling of the asteroids, tidal deformations affecting the gravitational potential of Earth and Moon, modelling of the lunar core and unstable delay offsets for the calibration. These errors are different for each observation. Random errors result from the general measurement accuracy of LLR. They are different for each night and depend on the observatory. To assess whether such a scaling factor is necessary when estimating ERPs from LLR, a sensitivity analysis was carried out by creating variations in the fitted and fixed parameters to obtain multiple solutions. The fitted ERPs from different calculations were then compared to each other. Four cases were run for the different calculations:
-
1.
Case 1.1: Initial values of all parameters (including the velocities of the LLR observatories) from a standard solution of LUNAR. All standard parameters along with the ERPs for selected nights were fitted.
-
2.
Case 1.2: Similar to case 1.1, except only ERPs on selected nights were fitted and the standard parameters were kept fixed.
-
3.
Case 2.1: Initial values of all parameters from a solution of LUNAR which was obtained by fixing the velocities of the LLR observatories to ITRF2020 values. All standard parameters except the velocities of the LLR observatories were fitted, along with the ERPs for selected nights.
-
4.
Case 2.2: Similar to case 2.1, except only ERPs on selected nights were fitted and the standard parameters (including the velocities of the LLR observatories) were kept fixed.
For the sensitivity analysis, case 2.1 was selected as standard case against which the results of the other cases are compared. This case was taken because the specific LLR network of the observatories, determined from the LLR analysis, is stabilised by the fixed ITRF velocities.
The ERPs for the sensitivity analysis were determined from the NPs of all LLR observatories. The minimum number of NPs was 15 which results in 491 nights in the time span 04.1984–03.2022. Each component of the ERPs, that is \(x_p\), \(y_p\), and \(\Delta \)UT1, was determined in a separate adjustment procedure. The IERS C04 series was used as the a-priori ERP. Its values have been fixed for the nights not considered in the fit, which helps to keep the LLR internal network closer to the ITRF. Studies with more LLR data sets are discussed in Singh (2023).
The results of the sensitivity analysis are given in Table 1. All values in the table are the WRMS, weighted according to the number of NPs per night. Column three of Table 1 shows the formal errors of the LSA, without a scaling factor (1\(\sigma \) (2.1)). This means, from the individual 491 formal errors for the determined ERP component, the WRMS is calculated, weighted according to the number of NPs per night. For column four, the mean of the formal errors of the cases 1.1, 1.2 and 2.2 is calculated for each night. This mean value is subtracted from case 2.1, and from all that differences, the WRMS is calculated and given in column four. Finally, for the result in column five, the standard deviation of the three cases 1.1, 1.2 and 2.2 (Std. Dev.) is calculated for each night, and from these values the WRMS is given in column five. For each estimated ERP, the results are split into two time spans, before and after 2000, as the results become much better after 2000 (Singh et al. 2022a).
For the two pole coordinates the results are very similar. The formal errors from the LSA are bigger than the differences between the standard case and the other cases, and also bigger than the standard deviation of the three cases. This is true for both time spans, before and after 2000. The result shows that a scaling factor for the formal errors of the adjustment is not necessary for the pole coordinates with the current version of the analysis model. The results for the Earth rotation phase \(\Delta \)UT1 differ from those of the pole coordinates. For the time span before 2000, only the difference between the standard case and the other cases is smaller than the formal errors of the adjustment. The standard deviations of the three cases are bigger than the formal errors of the adjustment. To ensure that the formal errors of the adjustment are bigger than the other values and represent a realistic uncertainty, at least a scaling factor of two is required here. Further investigations with more subsets show that a scaling factor of three is needed (Singh 2023). Also for the time span after 2000, the formal errors of the adjustment are only slightly bigger than the difference of the standard case to the other cases and the standard deviations of the other cases. So, a scaling factor of two should be used for \(\Delta \)UT1.
Figure 1 shows the results from the sensitivity analysis for case 2.1 as differences to the a-priori IERS C04 series and the uncertainties. The individual sub-figures show two aspects: (1) The large amount of data after 2015, because of the IR measurements provided by OCA, and (2) the improved uncertainties in the calculated ERP values, from 2015 onwards. The improved uncertainties are due to the high accuracy of the data. For the pole coordinates, the differences to the a-priori IERS C04 EOP series vary in the range of \(-\)6.0 mas to 5.1 mas for \(x_p\) and \(-\)4 mas to 3.7 mas for \(y_p\). The uncertainties (\(1\sigma \)) vary between 0.1 mas to 1.8 mas for \(x_p\) and 0.0 mas to 1.6 mas for \(y_p\). For \(\Delta \)UT1, the differences to the a-priori IERS C04 EOP series vary in the range of \(-\)266.2 \(\upmu \)s to 151.4 \(\upmu \)s. The uncertainties (\(2\sigma \)) vary between 0.1 \(\upmu \)s to 37.1\(\upmu \)s.
Results for the determination of ERPs from the NPs of all observatories. The left plots show the differences to the a-priori IERS C04 series and the right plots the uncertainties. For the pole coordinates the uncertainties are given as the formal errors of the LSA, for \(\Delta \)UT1 as two times the formal errors. (a) \(x_p\) differences. (b) \(\sigma (x_p)\) uncertainties. (c) \(y_p\) differences. (d) \(\sigma (y_p)\) uncertainties. (e) \(\Delta \)UT1 differences. (f) \(2\sigma \)(\(\Delta \)UT1) uncertainties
From the sensitivity analysis the best current uncertainty for the pole coordinates (1\(\sigma \)) is 0.47 mas for \(x_p\) and 0.59 mas for \(y_p\). Using the Earth radius at the equator as 6378 km, 1 mas corresponds to 3 cm spatial resolution on Earth’s surface. Thus, the uncertainty for the pole coordinates result in 1.41 cm and 1.77 cm spatial resolution, respectively. The current best uncertainty for \(\Delta \)UT1 is 12.36 \(\upmu \)s (2\(\sigma \)). Using the Earth radius, 10 \(\upmu \)s corresponds to 4.6 mm on the Earth’s surface and lead to a spatial resolution of 5.69 mm for \(\Delta \)UT1. As the ERP components were determined in separate adjustments the given uncertainties might be too optimistic because correlations between UT1 and the pole coordinates are not taken into account. In an earlier study by Singh et al. (2022a), the simultaneous determination of the two pole coordinates \(x_p\) and \(y_p\) has already been investigated. The uncertainties for the pole coordinates from a simultaneous determination were only slightly higher than from a separate determination with 15 NPs per night in the least-squares adjustment.
Compared to other space geodetic techniques like VLBI, Global Navigation Satellite System (GNSS) and Satellite Laser Ranging (SLR) the results from LLR are still worse. The uncertainties for \(\Delta \)UT1 from VLBI are about 3 \(\upmu \)s to 5 \(\upmu \)s from 24h sessions and 15 \(\upmu \)s to 20 \(\upmu \)s from intensive sessions. For \(x_p, y_p\) from VLBI, the uncertainties are about 50 \(\upmu \)as to 80 \(\upmu \)as (Schuh and Behrend 2012; Raut et al. 2022), about 10 \(\upmu \)as to 30 \(\upmu \)as from SLR (Sciarretta et al. 2010) and about 5 \(\upmu \)as to 20 \(\upmu \)as from GNSS (Capitaine 2017; Zajdel et al. 2020). Nevertheless, when the LLR results become better in the future, a possible contribution as an independent technique could be considered.
3.2 Nutation
As mentioned in Eq. (1), all EOPs are needed for the transformation of station coordinates from the ITRS into the inertial system. The IAU 2000 nutation model is described in the IERS Conventions 2010 (Petit and Luzum 2010) as a series for nutation in longitude \(\Delta \psi \) and obliquity \(\Delta \epsilon \), referred to the mean ecliptic of date:
with \(ARG = \sum _j^5 N_j F_j\), \(N_j\): multipliers, \(F_j\): Delaunay parameters and time t measured in Julian centuries from epoch J2000. n defines the number of terms the model is composed of, 678 lunisolar and 687 planetary terms with in-phase (first part of the sum in Eqs. (4) and (5)) and out-of-phase (second part of the sum) coefficients. This series is based on the REN2000 nutation solution (Souchay et al. 1999) for the rigid Earth, which is convolved to the nutation model MHB2000 for the non-rigid Earth by the transfer function from Mathews et al. (2002). This model is used as a-priori nutation model in the LLR analysis, where the non-time-dependent coefficients can be determined along with other parameters of the Earth-Moon system.
Nutation by definition refers to a dynamic reference system. In order to minimise possible small systematic deviations in orientation between the kinematic realisation of the inertial system based on VLBI and the inertial system which is dynamically realised by the LLR-based ephemeris computation, an additional perturbation rotation matrix is defined as
where \(\Theta _x\) is used to adjust the ecliptic angle, \(\Theta _y\) allows an adjustment of the GCRS equator, \(\Theta _z\) an adjustment between the vernal equinox and the origin along the equator. The perturbation rotations are modelled as time-dependent quantities with \(\Theta = \Theta _0 + \dot {\Theta } \Delta t\). A similar approach to fitting the reference systems is also described in Hilton and Hohenkerk (2004), Yagudina (2009), Zerhouni and Capitaine (2009), Williams et al. (2013). In a first step of the analysis, the angles \(\Theta _x\) and \(\Theta _y\) were determined with the fixed nutation model to minimise the deviations between the reference systems. In a second step, the components of Eq. (6) were fixed to determine the coefficients of the nutation series.
Table 2 gives the values from the LLR analysis for the periods with the largest contribution to the nutation angles. These periods are: 18.6-year, 182.62-day, 13.66-day, 9.3-year, and 365.26-day, sorted in order of their largest contribution. The values are given as differences to the a-priori model with uncertainties as three times the formal errors from the LSA. The current results are compared to those from Hofmann et al. (2018), where a shorter time span of NPs was used, in particular fewer NPs measured in IR were used. Looking at the differences to the a-priori model, the 2022 results are smaller than the 2018 results in most cases, and the uncertainties have improved by a factor of two. The largest improvement is for the 13.66-day period, where the benefit from IR OCA data and the associated more homogeneous observation of the lunar orbit is clearly visible. The uncertainties are still the formal errors with a scaling factor of three to be comparable with the 2018 results. In future, a sensitivity analysis similar to the ERPs will also be carried out for the determination of the nutation coefficients from the LLR analysis in order to assess the need for such a scaling factor.
4 Conclusions
A 52-year LLR data set has been analysed to determine EOPs. For the determination of the terrestrial pole coordinates and the Earth rotation phase a sensitivity analysis was performed in order to assess the need for a scaling factor of the formal errors from the LSA. Different cases and time periods were investigated. For the terrestrial pole coordinates, a scaling factor is not needed. However, for the Earth rotation phase, a scaling factor of two (after 2000) seems to be reasonable. The current best results are 0.47 mas for \(x_p\), 0.59 mas for \(y_p\) and 12.36 \(\upmu \)s with a scaling factor of two for \(\Delta \)UT1. Nevertheless, the LLR uncertainties might be too optimistic because correlations between UT1 and the polar coordinates are not taken into account when determining the ERPs components separately. Therefore as next step, UT1 and the pole coordinates will be determined together and analysed to find the best strategy for ERP determination from LLR data. It will also be further investigated, which parameters of the Earth-Moon system should be determined together with the ERPs. This will lead to a more realistic estimation of their uncertainties.
Compared to results for the nutation coefficients from the year 2018, the current differences to the a-priori MBH2000 model are smaller in most cases, and the uncertainties have improved by a factor of two. Here, the high number of IR NPs and the more homogeneous tracking of the lunar orbit are beneficial, especially for the 13.66-day nutation period.
With more IR data from the observatories OCA and WLRS, it is expected that the parameters of the LSA and also the EOPs can be further improved. Compared to other space geodetic techniques, the results from LLR still lag behind. However, the results are still important as LLR is the only technique other than VLBI which can provide \(\Delta \)UT1 and nutation values with some good accuracy, and therefore can be used to verify the VLBI results. In future a combined analysis of LLR and VLBI data for the EOPs determination is planned.
References
Biskupek L (2015) Bestimmung der Erdorientierung mit Lunar Laser Ranging. PhD thesis, Leibniz Universität Hannover. https://doi.org/10.15488/4721
Biskupek L, Müller J, Hofmann F (2012) Determination of nutation coefficients from lunar laser ranging. In: Kenyon S, Pacino MC, Marti U, et al (eds) Geodesy for Planet Earth, International Association of Geodesy Symposia, vol 136. Springer, Berlin, Heidelberg, pp 521–525. https://doi.org/10.1007/978-3-642-20338-1_63
Biskupek L, Müller J, Torre JM (2021) Benefit of new high-precision LLR data for the determination of relativistic parameters. Universe 7(2). https://doi.org/10.3390/universe7020034
Biskupek L, Singh VV, Müller J (2022) Estimation of earth rotation parameter UT1 from lunar laser ranging observations. Springer, Berlin, Heidelberg, pp 1–7. International Association of Geodesy Symposia. https://doi.org/10.1007/1345_2022_178
Bizouard C, Lambert S, Gattano C, et al (2019) The IERS EOP 14C04 solution for Earth orientation parameters consistent with ITRF 2014. J Geodesy 93(5):621–633. https://doi.org/10.1007/s00190-018-1186-3
Capitaine N (2017) The determination of earth orientation by VLBI and GNSS: principles and results. In: Arias EF, Combrinck L, Gabor P, et al (eds) The science of time 2016. Springer International Publishing, Cham, pp 167–196. https://doi.org/10.1007/978-3-319-59909-0_23
Chabé J, Courde C, Torre JM, et al (2020) Recent progress in lunar laser ranging at grasse laser ranging station. Earth Space Sci 7(3):e2019EA000,785. https://doi.org/10.1029/2019EA000785
Chapront-Touzé M, Chapront J, Francou G (2000) Determination of UT0 with LLR observations. In: Proceedings of the Journées 1999 “Motion of Celestial Bodies, Astrometry and Astronomical Reference Frames”, pp 217–220
Cheng YT, Liu JC, Zhu Z (2019) Analyses of celestial pole offsets with VLBI, LLR, and optical observations. Astron Astrophys 627:A81. https://doi.org/10.1051/0004-6361/201834785
Dickey JO, Newhall XX, Williams JG (1985) Earth orientation from lunar laser ranging and an error analysis of polar motion services. J Geophys Res 90(B11):9353–9362
Eckl JJ, Schreiber KU, Schüler T (2019) Lunar laser ranging utilizing a highly efficient solid-state detector in the near-IR. In: Prochazka I, Sobolewski R, James RB, et al (eds) Quantum Optics and Photon Counting 2019, International Society for Optics and Photonics, vol 11027. SPIE, p 1102708. https://doi.org/10.1117/12.2521133
Fukushima T (2003) A new precession formula. Astron J 126(1):494–534. http://stacks.iop.org/1538-3881/126/i=1/a=494
Hilton JL, Hohenkerk CY (2004) Rotation matrix from the mean dynamical equator and equinox at J2000.0 to the ICRS. Astron Astrophys 413(2):765–770. https://doi.org/10.1051/0004-6361:20031552
Hofmann F, Müller J (2018) Relativistic tests with lunar laser ranging. Classic Quant Gravity 35(3):035,015. https://doi.org/10.1088/1361-6382/aa8f7a
Hofmann F, Biskupek L, Müller J (2018) Contributions to reference systems from Lunar Laser Ranging using the IfE analysis model. J Geodesy 92(9):975–987. https://doi.org/10.1007/s00190-018-1109-3
Mathews PM, Herring TA, Buffett BA (2002) Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth’s interior. J Geophys Res 107(B4):ETG 3–1–ETG 3–30. https://doi.org/https://doi.org/10.1029/2001JB000390
Michelsen EL (2010) Normal point generation and first photon bias correction in APOLLO Lunar Laser Ranging. PhD thesis, University of California, San Diego
Müller J (1991) Analyse von Lasermessungen zum Mond im Rahmen einer post-Newton’schen Theorie. PhD thesis, Technische Universität München, Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften, Reihe C, Nr. 383
Müller J, Biskupek L, Oberst J, et al (2009) Contribution of lunar laser ranging to realise geodetic reference systems. In: Drewes H (ed) Geodetic Reference Frames, International Association of Geodesy Symposia, vol 134. Springer, Berlin, Heidelberg, pp 55–59. https://doi.org/10.1007/978-3-642-00860-3_8
Noll CE (2010) The crustal dynamics data information system: A resource to support scientific analysis using space geodesy. Adv Space Res 45(12):1421–1440. https://doi.org/10.1016/j.asr.2010.01.018
Park RS, Folkner WM, Williams JG, et al (2021) The JPL planetary and lunar ephemerides DE440 and DE441. Astron J 161(3):105. https://doi.org/10.3847/1538-3881/abd414
Pavlov D (2019) Role of lunar laser ranging in realization of terrestrial, lunar, and ephemeris reference frames. J Geodesy 94(1):5. https://doi.org/10.1007/s00190-019-01333-y
Pavlov DA, Williams JG, Suvorkin VV (2016) Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models. Celest Mech Dyn Astron 126(1):61–88. https://doi.org/10.1007/s10569-016-9712-1
Pearlman MR, Noll CE, Pavlis EC, et al (2019) The ILRS: approaching 20 years and planning for the future. J Geodesy 93(11):2161–2180. https://doi.org/10.1007/s00190-019-01241-1
Petit G, Luzum B (eds) (2010) IERS Conventions 2010. No. 36 in IERS Technical Note, Verlag des Bundesamtes für Kartographie und Geodäsie, Frankfurt am Main
Ratcliff JT, Gross RS (2020) Combinations of earth orientationmeasurements: SPACE2019, COMB2019,and POLE2019. Tech. Rep. JPL Publication 20-3
Raut S, Heinkelmann R, Modiri S, et al (2022) Inter-comparison of UT1-UTC from 24-hour, intensives, and VGOS sessions during CONT17. Sensors 22(7). https://doi.org/10.3390/s22072740
Schuh H, Behrend D (2012) VLBI: A fascinating technique for geodesy and astrometry. J Geodynam 61:68–80. https://doi.org/https://doi.org/10.1016/j.jog.2012.07.007
Sciarretta C, Luceri V, Pavlis EC, et al (2010) The ILRS EOP time series. Artif Satell 45:41–48. https://doi.org/10.2478/v10018-010-0004-9
Singh VV (2023) Lunar laser ranging - improved modelling and parameter estimation. PhD thesis, Leibniz Universität Hannover. https://doi.org/10.15488/14298
Singh VV, Biskupek L, Müller J, et al (2022a) Earth rotation parameter estimation from LLR. Adv Space Res 70(8):2383–2398. https://doi.org/10.1016/j.asr.2022.07.038
Singh VV, Biskupek L, Müller J, et al (2022b) Estimation of lunar ephemeris from lunar laser ranging. EGU abstract. https://doi.org/10.5194/egusphere-egu22-2815, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-2815
Souchay J, Loysel B, Kinoshita H, et al (1999) Corrections and new developments in rigid earth nutation theory - III. Final tables “REN-2000” including crossed-nutation and spin-orbit coupling effects. Astron Astrophys Suppl Ser 135:111–131. https://doi.org/10.1051/aas:1999446
Viswanathan V, Fienga A, Minazzoli O, et al (2018) The new lunar ephemeris INPOP17a and its application to fundamental physics. Month Not Roy Astrono Soc 476(2):1877–1888. https://doi.org/10.1093/mnras/sty096
Williams JG (1994) Contributions to the Earth’s obliquity rate, precession, and nutation. Astron J 108(2):711–724. https://doi.org/10.1086/117108
Williams JG, Turyshev SG, Boggs DH (2012) Lunar laser ranging tests of the equivalence principle. Class Quant Grav 29(18):184,004. https://doi.org/10.1088/0264-9381/29/18/184004
Williams JG, Boggs DH, Folkner WM (2013) DE430 lunar orbit, physical librations, and surface coordinates. JPL Interoffice Memorandum IOM 335-JW,DB,WF-20080314-001, Jet Propulsion Laboratory, California Institute of Technology, Passadena, California
Yagudina EI (2009) Lunar numerical theory EPM2008 from analysis of LLR data. In: Soffel M, Capitaine N (eds) Astrometry, Geodynamics and Astronomical Reference Systems, Lohrmann-Observatorium and Observatoire de Paris, Proceedings of the Journées 2008 Systèmes de Référence Spatio-Temporels, pp 61–64
Zajdel R, Sośnica K, Bury G, et al (2020) System-specific systematic errors in earth rotation parameters derived from GPS, GLONASS, and Galileo. GPS Sol 24(3):74. https://doi.org/10.1007/s10291-020-00989-w
Zerhouni W, Capitaine N (2009) Celestial pole offsets from lunar laser ranging and comparison with VLBI. Astron Astrophys 507(3):1687–1695. https://doi.org/10.1051/0004-6361/200912644
Zhang M, Müller J, Biskupek L (2020) Test of the equivalence principle for galaxy’s dark matter by lunar laser ranging. Celest Mech Dyn Astron 132(4):25. https://doi.org/10.1007/s10569-020-09964-6
Acknowledgements
We acknowledge with gratitude, that more than 53 years of processed LLR data have been obtained under the efforts of the personnel at McDonald Laser Ranging Station, USA, Côte d’Azur Observatory, France, Lure Observatory on Maui/Hawaii, USA, Apache Point Observatory Lunar Laser ranging Operation, USA, Matera Laser Ranging Observatory, Italy and Geodetic Observatory Wettzell, Germany. We also acknowledge with thanks the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC-2123 QuantumFrontiers - Project-ID 390837967).
Declarations
• Availability of data: LLR data is collected, archived, and distributed under the auspices of the International Laser Ranging Service (ILRS) (Pearlman et al. 2019). All LLR NPs used for these studies are available from the Crustal Dynamics Data Information System (CDDIS) at NASA’s Archive for Space Geodesy Data, USA, (Noll 2010) at the website.Footnote 1 The KEOF COMB2019 EOP time series is available at the websiteFootnote 2 and the IERS C04 EOP time series is available at the website.Footnote 3
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2024 The Author(s)
About this paper
Cite this paper
Biskupek, L., Singh, V.V., Müller, J., Zhang, M. (2024). Potential of Lunar Laser Ranging for the Determination of Earth Orientation Parameters. In: Freymueller, J.T., Sánchez, L. (eds) Gravity, Positioning and Reference Frames. REFAG 2022. International Association of Geodesy Symposia, vol 156. Springer, Cham. https://doi.org/10.1007/1345_2024_238
Download citation
DOI: https://doi.org/10.1007/1345_2024_238
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-63854-1
Online ISBN: 978-3-031-63855-8
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)