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 (Muller 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. 201). 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

$$\displaystyle \begin{aligned} {\mathbf{r}}_{GCRS} = \mathbf{Q}(dt)\: \mathbf{R}(dt)\: \mathbf{W}(dt) \ {\mathbf{r}}_{ITRS} \,. {} \end{aligned} $$
(1)

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 ERP from LRR data is given by Dickey et al. (1985), Muller (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

$$\displaystyle \begin{aligned} \Delta \text{UT}0 = \Delta \text{UT}1 + \frac{(x_p \sin{}(\lambda) + y_p \cos{}(\lambda)) \tan{}(\phi)}{15 \times 1.002737909}\ , {} \end{aligned} $$
(2)

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

$$\displaystyle \begin{aligned} \text{VOL} = x_p \cos \lambda - y_p \sin \lambda \ . {} \end{aligned} $$
(3)

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 (Muller 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. 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. 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. 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. 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).

Table 1 The values in each column are the weighted root mean square (WRMS) values, weighted according to the number of NPs per night. The last three columns show the formal errors of the LSA (1\(\sigma \) values) for the standard case 2.1, the difference of standard case 2.1 to the mean of the other cases (MC) and the standard deviation (Std. Dev.) of the cases 1.1, 1.2 and 2.2. For each estimated ERP, the results are split into two time spans, before and after 2000
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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 \(\upmu \)m to 151.4 \(\upmu \)s. The uncertainties (\(2\sigma \)) vary between 0.1 \(\upmu \)s to 37.1\(\upmu \)s.

Fig. 1
figure 1

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

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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 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 EOP 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:

$$\displaystyle \begin{aligned} \Delta \psi = \sum_{i=1}^{n} (A_i + A^{\prime}_i t) \ \sin (ARG) + (A^{\prime\prime}_i + A^{\prime\prime\prime}_i t) \ \cos (ARG) {} \end{aligned} $$
(4)
$$\displaystyle \begin{aligned} \Delta \epsilon = \sum_{i=1}^{n} (B_i + B^{\prime}_i t) \ \cos (ARG) + (B^{\prime\prime}_i + B^{\prime\prime\prime}_i t) \ \sin (ARG) {} \end{aligned} $$
(5)

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

$$\displaystyle \begin{aligned} {} \mathbf{S}(dt) = \begin{bmatrix} 1 & \Theta_z & -\Theta_y \\ -\Theta_z & 1 & \Theta_x \\ \Theta_y & -\Theta_x & 1 \end{bmatrix} \end{aligned} $$
(6)

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.

Table 2 Differences between the a-priori MHB2000 nutation model and values determined from LLR data for nutation coefficients of different periods. The values are given with the uncertainties as three times the formal errors from the LSA. The results from 2018 are compared to the results from 2022. The values are given in mas
Full size table

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 NP 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.