TY - JOUR
AU1 - Kurdubov, Sergei, L
AU2 - Pavlov, Dmitry, A
AU3 - Mironova, Svetlana, M
AU4 - Kaplev, Sergey, A
AB - ABSTRACT Modern radio astrometry has reached the limit of the resolution that is determined by the size of the Earth. The only way to overcome this limit is to create radio telescopes outside our planet. There is a proposal to build an autonomous remote-controlled radio observatory on the Moon. Working together with existing radio telescopes on Earth in the very-long-baseline interferometry (VLBI) mode, the new observatory will form an interferometer baseline up to 410 000 km, enhancing the present astrometric and geodetic capabilities of VLBI. We perform numerical simulations of Earth–Moon VLBI observations, operating simultaneously with the international VLBI network. It is shown that these observations will significantly improve the precision of the determination of the Moon’s orbital motion, libration angles, the International Celestial Reference Frame and relativistic parameters. relativistic processes, instrumentation: interferometers, methods: numerical, ephemerides, reference systems, Moon 1 INTRODUCTION AND HISTORY Very-long-baseline interferometry (VLBI) observations allow us to determine various parameters that are important for astrometry or geodesy, such as the coordinates of extragalactic radio sources, Earth’s rotation parameters and the coordinates of stations, with an accuracy proportional to λ/b, where λ is the wavelength and b is the baseline between two radio telescopes. At present, the wavelength is 7.5 mm or more, while the baseline is bounded by the diameter of the Earth. Expansion towards shorter wavelengths is difficult because of atmospheric absorption, the instability of frequency standards and the limitations of data acquisition systems. Hence, the most straightforward way to improve accuracy is the extension of the baseline outside Earth. Placing one of the interferometer antennae on the Moon will allow us to increase the baseline by a factor of 60 for the international VLBI network. The Moon has served as a platform for scientific experiments since Project Apollo. The most important experiments concerning lunar dynamics and selenodesy include lunar laser ranging to five lunar retroreflectors (Apollo 11, 14 and 15, and Luna 17 and 21), the Gravity Recovery and Interior Laboratory (GRAIL) experiment, which measured the lunar gravitational field with unprecedented accuracy, and the Lunar Reconnaissance Orbiter (LRO) spacecraft that provides high-resolution mapping and altimetry. The lunar lander Chang'e 3, which landed at the end of 2013, is operational to this day, as is the LRO. It is evident that space agencies of different countries are headed towards lunar exploration and perhaps a habitable lunar base in the near future. The far side of the Moon is of particular interest for astrophysical experiments because it is naturally shielded from Earth’s natural and artificial radio frequency interference (Douglas & Smith 1985; Lazio et al. 2009; Skalsky et al. 2014). One of the earliest proposals for Moon–Earth radio interferometry was made by Burns (1985, 1988). To this day, no attempt of such an experiment has been made; however, two orbital VLBI telescopes have been built. The first, the 8-m Highly Advanced Laboratory for Communications and Astronomy (HALCA; Hirabayashi et al. 2000), also known by its programme name, the VLBI Space Observatory Programme(VSOP), provided VLBI observations from 1997 to 2003. The second, the 10-m RadioAstron (Kardashev et al. 2013), also known as Spectr-R, launched in 2011 and is still operational. However, while both HALCA and RadioAstron have provided great observations for astrophysics, neither has been used for the purposes of astrometry. One reason for this is that it is difficult to track the spacecraft orbit precisely during the VLBI session; this is essential for building the reference frame but not so essential for obtaining images and studying the structure of extragalactic radio sources. One possible difficulty in VLBI on such a large baseline could be the loss of correlated flux densities. However, the Spectr-R results show that at least 160 quasars have very compact structures and high brightness temperatures, enough to obtain fringes at baselines up to 200 000 km (Kardashev et al. 2017). Modern VLBI Global Observing System (VGOS) recording systems can register more than 100× wider bandwidth than the Spectr-R formatter (16 MHz) so we can expect that the new instrument will have more than 10× sensitivity and that fringes will be obtained. 2 CONCEPT 2.1 Pros and cons of lunar environment There are great challenges involved in placing a moving instrument on the lunar surface, such as complicated landing and maintenance, temperature jumps and solar radiation. However, in some ways, the Moon provides better experimental conditions than Earth. Having no atmosphere and no oceans, the Moon has a much more stable rotational motion than the Earth; also, the absence of an atmosphere eases the processing of observations and reduces errors that come from the ionosphere and troposphere interfering with observed radio noise. Low lunar gravity will allow lighter construction and machinery. A spacecraft orbiting the Earth is also not affected by the atmosphere; however, the trajectory of a spacecraft is highly unstable compared with an object tied to lunar surface. The biggest source of instability is solar pressure, which does affect the Moon, but its effect on the Moon is much more subtle, predictable and stable (Vokrouhlický 1997). Furthermore, the spacecraft orbit always will be a byproduct of astrometric observations, whereas the precise positioning of a lunar-based antenna contributes significantly to selenodesy science. In order to participate in modern astrometric and geodetic VLBI observations, an orbital VLBI telescope has to change targets fast. Modern Earth-based VGOS antennas have slewing speeds of up to 6 deg s–1. The corresponding rotation of spacecraft in space can be achieved only with reactive thrusters, leading to a very unstable orbit. Also, the fuel tanks will limit operational time. 2.2 Co-location It is recommended that the new radio telescope is co-located with a retroreflector, either a panel of corner cubes (Vasiliev et al. 2014), or a next-generation single cube retroreflector (Turyshev et al. 2013; Araki et al. 2016). Because it requires no power on the Moon and no data transmission, the Earth–Moon laser ranging would help to determine the precise location of the station. Also, it would independently contribute to the study of lunar dynamics, the building of the lunar reference frame and testing of general relativity (Murphy 2013; Williams, Boggs & Ratcliff 2013b; Pavlov, Williams & Suvorkin 2016; Viswanathan et al. 2018). The addition of a global navigation satellite system (GNSS) receiver to the new lunar VLBI station will allow it to receive signals from Earth’s satellites, when the Moon happens to be in the same beam as Earth when viewed from the satellite. Such observations would allow us to improve the determination of the orbits of GNSS satellites in the celestial reference frame. 2.3 Choice of location Only the scientific outcome was considered as the primary criterion for the location of the proposed station on the lunar surface. Other criteria were not considered, such as landing and deployment issues, local terrain, power supply and proximity to possible lunar baselines. There are multiple reasons for the location to be chosen on the visible side of the lunar disc. First, this would allow direct Moon–Earth data transfer. Secondly, it would allow co-location with a lunar laser ranging (LLR) retroreflector and/or GNSS receiver (see Section 2.2). Thirdly, it would allow a slight (roughly 9000 km) increase in the interferometry baseline, by observing the radio sources that are visible from the far side of the Earth (as viewed from the Moon). The location on the visible side of the Moon exposes the station to radio interference from the Earth (natural or artificial). However, the receiving pattern of the lunar radio telescope will be narrow, similarly to its Earth VLBI counterparts, so the radio interference will not be a problem. Assuming there is only one instrument of this kind, it should be located close to the lunar equator so that the radio sources in both the southern and nothern hemispheres are visible. The LLR (currently the most precise tool to study lunar physical librations) has low sensitivity to the rotation of the Moon around the Moon–Earth direction (see Fig. 1). The new VLBI station can achieve this sensitivity if it is placed outside the central area of the lunar disc. Another benefit of a location outside the central area is that the new station would extend the current lunar reference frame, whose accuracy currently deteriorates outside the central area in which the five retroreflector points are located. Figure 1. Open in new tabDownload slide Five existing lunar retroreflector panels and the proposed VLBI station. The XY plane and the Z-axis are close to the lunar equator and rotation axis, respectively. The X-axis is directed approximately towards the Earth. Figure 1. Open in new tabDownload slide Five existing lunar retroreflector panels and the proposed VLBI station. The XY plane and the Z-axis are close to the lunar equator and rotation axis, respectively. The X-axis is directed approximately towards the Earth. Here, the chosen location is on the visible part of the Moon’s equator, near its western end. The combination of the existing retroreflectors and the new VLBI instrument will allow us to determine all three ‘instant’ (daily/weekly/monthly) rotational corrections to the lunar orientation. LLR or VLBI alone are able to determine two rotations each. 3 DECISIONS ABOUT THE LUNAR MODEL 3.1 Lunar dynamical model and ephemeris We use the dynamical model of the orbital and rotational motion of the Moon implemented within the Ephemerides of Planets and the Moon (EPM; Pavlov et al. 2016). The rotational part of the model, involving tidal and rotational dissipation, as well as a spherical liquid core, was proposed by Williams et al. (2001). Later improvements, such as core flattening and dedicated parameters describing the influence of Earth’s tides on the orbit of the Moon, were implemented for the DE430 ephemeris (Folkner et al. 2014). The third publicly available lunar ephemeris, INPOP, is presently based on the same model (Viswanathan et al. 2018). Lunar ephemerides contain the geocentric (i.e. the geocentric celestial reference frame) position r and velocity |$\dot{\mathbf {r}}$| of the Moon as functions of time, and also the Euler angles ϕ, θ and ψ and their rates as functions of time. The lunar frame is aligned with the principal axes of the undistorted lunar mantle. The Euler angles define the matrix of the transformation from the lunar frame to the celestial frame: \begin{eqnarray*} R_{\mathrm{L2C}}(t)=R_z[\phi (t)]R_x[\theta (t)]R_z[\psi (t)], \end{eqnarray*} (1) whereRx and Rz are matrices of right-hand rotations of vectors around axes x and z, respectively. The argument t will be omitted when appropriate. 3.2 Daily lunar orientation parameters We study whether it is possible to determine daily corrections to the dynamical model of transformation from the lunar frame to the celestial frame: \begin{eqnarray*} R_{\mathrm{L2C}}^{\textrm{daily}}(t)=R_x(\Delta X) R_y(\Delta Y) R_z(\Delta Z) R_{\mathrm{L2C}}(t) . \end{eqnarray*} (2) The daily corrections are assumed to be small and the order of the X, Y and Z rotations is not relevant. 3.3 Determination of the reference point of the radio telescope The VLBI technique requires precise determination of the position of the lunar radio telescope with respect to (w.r.t.) its Earth counterparts. Such a determination can be done in two different ways depending on the task. If the task is to build the lunar ephemeris and/or lunar reference frame, the selenocentric position |${\boldsymbol l}_\mathrm{PA}$| of the lunar radio telescope should be fitted to observations. The VLBI reduction routine should use the determined |${\boldsymbol l}_\mathrm{PA}$| to obtain the GCRS position |${\boldsymbol l}_\mathrm{GCRS}$|⁠: \begin{eqnarray*} {\boldsymbol l}_\mathrm{GCRS}(t) &=& \boldsymbol r(t) + {\boldsymbol l}_\mathrm{LCRS}(t)\left[1-\frac{U(t)}{c^2}\right] - \frac{{\boldsymbol {\dot{r}}}_{\mathrm{B}}(t)\cdot {\boldsymbol l}_\mathrm{LCRS}(t)}{2c^2}{\boldsymbol {\dot{r}}_{\mathrm{B}}(t)}; \nonumber\\ {\boldsymbol l}_\mathrm{LCRS}(t) &=& R_\textrm{L2C}(t)\, {\boldsymbol l}_\mathrm{PA}+ \boldsymbol \Delta ({\boldsymbol l}_\mathrm{PA},t). \end{eqnarray*} (3) Here, U is the gravitational potential at the Moon’s centre, excluding the Moon’s mass, |$\boldsymbol {r}_{\mathrm{B}}$| and |$\boldsymbol {\dot{r}}_{\mathrm{B}}$| are the barycentric position and velocity of the Moon, respectively, and |$\boldsymbol \Delta$| is the displacement due to the solid Moon tide raised by Earth and the Sun. For the purposes of astrometry, it is more natural to determine the geocentric, rather than selenocentric, position of the lunar radio telescope. This would justify fixing – not determining – the lunar ephemeris together with the whole Earth–Moon VLBI solution. The link between the lunar ephemeris and the geocentric position of the retroreflector is relatively small. The VLBI reduction should then use the determined |${\boldsymbol l}_\mathrm{GCRS}(t_0)$| to solve for |${\boldsymbol l}_\mathrm{LCRS}(t_0)$|⁠, then solve for |${\boldsymbol l}_\mathrm{PA}$| and, finally, calculate |${\boldsymbol l}_\mathrm{GCRS}(t)$| using equation (3). To estimate the accuracy of the determination of |${\boldsymbol l}_\mathrm{GCRS}(t_0)$|⁠, the relativistic and tidal terms can be neglected and the derivative matrix of |${\boldsymbol l}_\mathrm{GCRS}(t)$| w.r.t. |${\boldsymbol l}_\mathrm{GCRS}(t_0)$| will be simplified to \begin{eqnarray*} \frac{{\mathrm{d}} {\boldsymbol {\boldsymbol l}_\mathrm{GCRS}(t)}}{{\mathrm{d}} {\boldsymbol {\boldsymbol l}_\mathrm{GCRS}(t_0)}} \approx R_\textrm{L2C}(t)R_\textrm{L2C}^T(t_0) . \end{eqnarray*} (4) 4 SETTING OF EXPERIMENT 4.1 Data We simulated a two-week intensive Earth–Moon VLBI campaign similar to a subset of CONT17 (Nothnagel, Artz & Behrend 2017).1 The Legacy-1 subset was created; it has 14 VLBI stations in Europe, Russia, South Africa, Australia, New Zealand, Brazil, Japan and Hawaii. To each scan of an Earth radio telescope observing a source, we add a set of Earth–Moon VLBI delays of the same source for each of the Earth radio telescopes participating in the session. Scans involving a quasar not visible from the lunar radio telescope at the specified time were excluded. The final data set contains 24 095 Earth–Moon simulated VLBI delays and 105 808 real delays on Earth–Earth baselines for 9 d. The whole campaign lasted 15 d (from 2017 November 28 to December 12), but observations from the remaining 6 d of the Legacy-1 subset were not available at the time of writing. Of the three CONT17 subsets (i.e. Legacy-1, Legacy-2 and VGOS-Demo), the first subset has the best expected accuracy for the determination of the Earth orientation parameter (EOP). For comparison, we also took the Legacy-2 subset in the simulation of the determination of the lunar orientation parameter (LOP) and the parametrized post-Newtonian (PPN) parameter γ (see Sections 4.5 and 5.4). For this subset, 36 589 Earth–Moon delays were generated and used. Simulations of Earth–Moon VLBI delays for different subsets were done separately, not simultaneously, due to the time overlap. For one part of the experiment (see Section 4.3), we used the real LLR data spanning from 1970 to the end of 2017. The most important observations at present are performed at Apache Point Observatory (Murphy et al. 2012; Murphy 2013) and the Observatoire de la Côte d’Azur (Samain et al. 1998; Courde et al. 2017). More information about re-weighting and reductions of LLR observations is given in Pavlov et al. (2016). Some observations were considered erroneous and were filtered out. Most often, the observations from the McDonald Laser Ranging Station (1988–2015) did not fit well into the model. In total, of 25 535 LLR observations, 773 were filtered out and 24 762 were used in the lunar solution. 4.2 Software Two independent software packages were used. era-8 (Pavlov & Skripnichenko 2015) was used to fit the parameters of the lunar model to observations, and to integrate lunar ephemeris (Section 4.3). An extension was made to era-8 to process Earth–Moon VLBI delays (in addition to LLR normal points) as part of a global lunar solution. quasar (Gubanov & Kurdubov 2014) was used to obtain the VLBI solution for the celestial reference frame, using a fixed lunar ephemeris. An extension was made to quasar to bring a VLBI station to the Moon, to determine its location (see Section 4.4) or the LOP (see Section 4.5). 4.3 Estimating the improvement of the lunar ephemeris Two estimations of the accuracy of the lunar ephemeris were obtained: the first using real LLR observations (1970–2017) and the second using LLR and 15 d of simulated VLBI observations at the end of 2017. The celestial and terrestrial VLBI frames were fixed. The formal error of the simulated Earth-VLBI observations was set to 7 mm (1σ), similar to the post-fit RMS discrepancy of the ionosphere-free combination for the global VLBI solution. Each of the determined parameters falls into one of the following two categories. Dynamical parameters include: the initial Euler angles of lunar physical libration and their rates; the initial GCRS position and velocity of the Moon; the initial core angular velocity; undistorted J2, C32, S32 and S33 of the Moon; ratios between undistorted main moments of inertia of the Moon; the lunar core flattening and friction coefficient; two Earth tidal delays; the GM of the Earth–Moon system. Reduction parameters include: positions and velocities of LLR stations; selenocentric (principal axes) positions of the lunar retroreflectors; Love number h2 of the Moon; 28 specific biases for different time intervals. The selenocentric lunar radio telescope position was also determined in the solution with 15 d of simulated VLBI observations. Details about the determined parameters are given in Pavlov et al. (2016), but note that, in this current paper, two changes have been made. The first is the aforementioned determination of J2. Not fixing J2 to a GRAIL-determined value brings more realistic uncertainties into the lunar solution. The second change is that the additional three kinematic terms of the longitude libration (with amplitudes in several mas) were not determined and are thus absorbed into the formal errors of other dynamical parameters. Parameters from both categories were determined simultaneously using the least-squares method, and their formal errors and covariance matrix were calculated. In addition to the dynamical parameters, the dynamical model also has constant undistorted Stokes coefficients of Earth’s gravitational potential (up to 6 deg), taken from the EGM2008 (Pavlis et al. 2012) solution, and the Moon’s gravitational potential (C30, C31, S31 and others of 4–6 deg) from the GL1200 (Goossens et al. 2016) solution. From the latter solution, the lunar k2 Love number is also used. The covariance matrix of the parameters under variation was formed as a submatrix of the lunar solution covariance matrix with excluded reduction parameters, to which rows and columns were added corresponding to the constants. The resulting matrix |$\mathbf {M}$| had size (n + k) × (n + k), where n = 28 was the number of the dynamical determined parameters and k = 83 was the number of constants. In the added rows and columns, squares of the formal errors of corresponding constants were assigned to the diagonal terms, while the off-diagonal terms were set to zero. The Monte Carlo simulation of the lunar ephemeris was done in the following way. The Cholesky decomposition of the covariance matrix was calculated: |$\mathbf {M} = \mathbf {L}^\mathrm{T}\mathbf {L}$|⁠, where |$\mathbf {L}$| is a lower triangular matrix. Each sample X for the simulation was obtained as a sample from the multidimensional normal distribution described by |$\mathbf {M}$|⁠: |$X = \mathbf {L}^\mathrm{T} Y$|⁠, where Y is a (n + k)-vector sample of independent normally distributed random variables with zero mean and unit variance. For each of the sampled X, a corresponding ephemeris of orbital and rotational motion of the Moon was obtained by numerical integration. The integrated dynamical model contained not only the Moon, but also the Sun, all planets, Pluto, Ceres, Pallas, Vesta, Iris and Bamberga. The parameters related to the non-lunar part of the model were not varied, because their influence on the lunar ephemeris is relatively small, and their influence on the uncertainty of the lunar ephemeris is negligibly small. The chosen epoch was 2018 January 2, and numerical integration was for 5 yr ahead (2018–2022). We studied the scatter of the determined initial parameters of the Moon (position and Euler angles) and the evolution of those parameters over time. Two different Monte Carlo simulations were done, one with only LLR observations and the other with LLR and simulated VLBI observations. For each of the two simulations, 1000 samples were generated. 4.4 Estimating the improvement of the celestial reference frame In order to estimate the accuracy of the celestial reference frame, we perform the solution over all CONT17 data with source right ascension and declination as global parameters. The tropospheric delay, clock synchronization parameters, LOP and EOP were estimated for each session. For the Earth baselines, we also use the model noise instead of real VLBI observations because the real VLBI data depend heavily on clock corrections and the determination procedure of those corrections should work with consistent |$O-C$| values. Because we do not have Earth–Moon VLBI data consistent with the real Earth VLBI clock, we have taken the correlator estimated noise for all observations. The noise was multiplied by a factor of 4, which is an ad hoc value that represents the clock, troposphere, etc., scatter from the real world; it was found by comparing parameter errors obtained in Earth VLBI processing with model noise and real VLBI data. 4.5 Estimating the determination of the LOP The estimation of the LOP was done in a similar manner to the celestial reference frame determination (Section 4.4). With the chosen location of the lunar radio telescope (Section 2.3), its position in the celestial reference frame will be sensitive to ΔX and ΔZ daily corrections but not to ΔY. The ΔX and ΔZ corrections were assumed to be independent and constant for each of the nine days of the Legacy-1 subset of the CONT17 campaign with simulated Earth–Moon VLBI data. For comparison, a similar simulation was done with the Legacy-2 subset spanning 15 consecutive days. Lunar laser ranging can also independently contribute to the determination of the LOP. However, Earth–Moon ranges are sensitive to Y and Zrotations but have almost no sensitivity to X rotations. As shown by Pavlov & Yagudina (2017), the formal errors of daily (nightly) ΔY and ΔZ obtained on real LLR data are 1–2 mas at best. 5 RESULTS OF PARAMETER ESTIMATION 5.1 Lunar ephemeris, lunar reference frame and Moon–Earth reference frame The positions of the lunar reference points were determined simultaneously with the parameters of orbit and physical libration of the Moon, and other parameters (see Section 4.3). Table 1 shows the accuracy of the lunar reference frame implemented by five retroreflector points (in the case of real LLR data) or five retroreflector points plus one radio telescope point (with real LLR and simulated VLBI data). We can see a 6–10× improvement in accuracy of the existing five points from the VLBI data. One of the key factors of such a major improvement is that the VLBI observations are omnidirectional (i.e. they measure the projections of the ‘Earth observatory–lunar radio telescope’ vector in all directions), and so they are sensitive to the orbital position of the Moon. LLR observations are sensitive to the Earth–Moon distance but not so much to the position of the Moon on orbit. Table 1. Uncertainty of the selenocentric reference points in two solutions. A = Apollo, L = Lunokhod and LRT = lunar radio telescope. Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 14.5 cm 0.5 cm A11 Y 19.4 cm 2.0 cm A11 Z 5.3 cm 1.4 cm L1 X 15.7 cm 0.5 cm L1 Y 13.9 cm 2.3 cm L1 Z 8.5 cm 4.8 cm A14 X 12.8 cm 0.5 cm A14 Y 19.9 cm 2.0 cm A14 Z 5.4 cm 1.4 cm A15 X 11.5 cm 0.4 cm A15 Y 18.8 cm 1.6 cm A15 Z 7.9 cm 3.5 cm L2 X 14.0 cm 0.4 cm L2 Y 16.5 cm 2.2 cm L2 Z 7.5 cm 3.5 cm LRT X N/A 1.5 mm LRT Y N/A 0.9 mm LRT Z N/A 0.6 mm Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 14.5 cm 0.5 cm A11 Y 19.4 cm 2.0 cm A11 Z 5.3 cm 1.4 cm L1 X 15.7 cm 0.5 cm L1 Y 13.9 cm 2.3 cm L1 Z 8.5 cm 4.8 cm A14 X 12.8 cm 0.5 cm A14 Y 19.9 cm 2.0 cm A14 Z 5.4 cm 1.4 cm A15 X 11.5 cm 0.4 cm A15 Y 18.8 cm 1.6 cm A15 Z 7.9 cm 3.5 cm L2 X 14.0 cm 0.4 cm L2 Y 16.5 cm 2.2 cm L2 Z 7.5 cm 3.5 cm LRT X N/A 1.5 mm LRT Y N/A 0.9 mm LRT Z N/A 0.6 mm Open in new tab Table 1. Uncertainty of the selenocentric reference points in two solutions. A = Apollo, L = Lunokhod and LRT = lunar radio telescope. Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 14.5 cm 0.5 cm A11 Y 19.4 cm 2.0 cm A11 Z 5.3 cm 1.4 cm L1 X 15.7 cm 0.5 cm L1 Y 13.9 cm 2.3 cm L1 Z 8.5 cm 4.8 cm A14 X 12.8 cm 0.5 cm A14 Y 19.9 cm 2.0 cm A14 Z 5.4 cm 1.4 cm A15 X 11.5 cm 0.4 cm A15 Y 18.8 cm 1.6 cm A15 Z 7.9 cm 3.5 cm L2 X 14.0 cm 0.4 cm L2 Y 16.5 cm 2.2 cm L2 Z 7.5 cm 3.5 cm LRT X N/A 1.5 mm LRT Y N/A 0.9 mm LRT Z N/A 0.6 mm Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 14.5 cm 0.5 cm A11 Y 19.4 cm 2.0 cm A11 Z 5.3 cm 1.4 cm L1 X 15.7 cm 0.5 cm L1 Y 13.9 cm 2.3 cm L1 Z 8.5 cm 4.8 cm A14 X 12.8 cm 0.5 cm A14 Y 19.9 cm 2.0 cm A14 Z 5.4 cm 1.4 cm A15 X 11.5 cm 0.4 cm A15 Y 18.8 cm 1.6 cm A15 Z 7.9 cm 3.5 cm L2 X 14.0 cm 0.4 cm L2 Y 16.5 cm 2.2 cm L2 Z 7.5 cm 3.5 cm LRT X N/A 1.5 mm LRT Y N/A 0.9 mm LRT Z N/A 0.6 mm Open in new tab Table 2 is similar to Table 1 but it shows the accuracy in the geocentric position at epoch (see Section 3.3). In the LLR-only solution, the geocentric positions of the retroreflector are determined to be ≈1.6× better than the selenocentric positions. It is known that the X coordinate of each lunar retroreflector panel strongly correlates with the GM of the Earth–Moon system and also with the semimajor axis of the Moon at epoch (see Williams, Boggs & Folkner 2013a). The correlation causes the selenocentric position of a retroreflector at epoch and the position of the Moon at epoch to be detected with less accuracy than the geocentric position of the retroreflector at epoch. Table 2. Uncertainty of the geocentric (at epoch 2018.01.02) reference points in two solutions. A = Apollo, L = Lunokhod and LRT = lunar radio telescope. Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 3.9 cm 2.0 cm A11 Y 2.9 cm 0.9 cm A11 Z 14.0 cm 1.3 cm L1 X 3.1 cm 2.2 cm L1 Y 3.5 cm 1.7 cm L1 Z 14.1 cm 4.4 cm A14 X 3.1 cm 1.9 cm A14 Y 3.2 cm 1.1 cm A14 Z 14.4 cm 1.4 cm A15 X 2.6 cm 1.4 cm A15 Y 3.8 cm 1.7 cm A15 Z 13.6 cm 3.2 cm L2 X 3.7 cm 2.2 cm L2 Y 3.2 cm 1.4 cm L2 Z 13.7 cm 3.2 cm LRT X N/A 0.9 mm LRT Y N/A 1.7 mm LRT Z N/A 1.0 mm Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 3.9 cm 2.0 cm A11 Y 2.9 cm 0.9 cm A11 Z 14.0 cm 1.3 cm L1 X 3.1 cm 2.2 cm L1 Y 3.5 cm 1.7 cm L1 Z 14.1 cm 4.4 cm A14 X 3.1 cm 1.9 cm A14 Y 3.2 cm 1.1 cm A14 Z 14.4 cm 1.4 cm A15 X 2.6 cm 1.4 cm A15 Y 3.8 cm 1.7 cm A15 Z 13.6 cm 3.2 cm L2 X 3.7 cm 2.2 cm L2 Y 3.2 cm 1.4 cm L2 Z 13.7 cm 3.2 cm LRT X N/A 0.9 mm LRT Y N/A 1.7 mm LRT Z N/A 1.0 mm Open in new tab Table 2. Uncertainty of the geocentric (at epoch 2018.01.02) reference points in two solutions. A = Apollo, L = Lunokhod and LRT = lunar radio telescope. Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 3.9 cm 2.0 cm A11 Y 2.9 cm 0.9 cm A11 Z 14.0 cm 1.3 cm L1 X 3.1 cm 2.2 cm L1 Y 3.5 cm 1.7 cm L1 Z 14.1 cm 4.4 cm A14 X 3.1 cm 1.9 cm A14 Y 3.2 cm 1.1 cm A14 Z 14.4 cm 1.4 cm A15 X 2.6 cm 1.4 cm A15 Y 3.8 cm 1.7 cm A15 Z 13.6 cm 3.2 cm L2 X 3.7 cm 2.2 cm L2 Y 3.2 cm 1.4 cm L2 Z 13.7 cm 3.2 cm LRT X N/A 0.9 mm LRT Y N/A 1.7 mm LRT Z N/A 1.0 mm Coordinate 3σ (LLR only) 3σ (LLR and VLBI) A11 X 3.9 cm 2.0 cm A11 Y 2.9 cm 0.9 cm A11 Z 14.0 cm 1.3 cm L1 X 3.1 cm 2.2 cm L1 Y 3.5 cm 1.7 cm L1 Z 14.1 cm 4.4 cm A14 X 3.1 cm 1.9 cm A14 Y 3.2 cm 1.1 cm A14 Z 14.4 cm 1.4 cm A15 X 2.6 cm 1.4 cm A15 Y 3.8 cm 1.7 cm A15 Z 13.6 cm 3.2 cm L2 X 3.7 cm 2.2 cm L2 Y 3.2 cm 1.4 cm L2 Z 13.7 cm 3.2 cm LRT X N/A 0.9 mm LRT Y N/A 1.7 mm LRT Z N/A 1.0 mm Open in new tab As for the ‘LLR+VLBI’ solution, the geocentric and selenocentric positions of the six lunar points are determined with similar accuracy overall, again due to the VLBI measuring in all directions. To estimate the accuracy of either the lunar or Moon–Earth reference frame not at epoch, but for some time into the future, we can look at Figs 2 and 3. Maximum and RMS deviations of sampled ephemeris (see Section 4.3) w.r.t. the nominal ephemeris are plotted. In 5 yr without new observations, the accuracy of the Moon’s orbital position determined via 47 yr or LLR degrades to 4.6 m (maximum); when LLR is combined with 9 d of Earth–Moon VLBI observations, the maximum error drops to 0.6 m. Similarly, the 5-yr maximum error of lunar physical libration is 1.3 m with LLR and 0.15 m with LLR+VLBI. Figure 2. Open in new tabDownload slide Estimation (maximum and RMS) of the uncertainty of the orbital position of the Moon for 5 yr, assuming no observations since 2018, using only LLR data or LLR data with simulated VLBI data. Figure 2. Open in new tabDownload slide Estimation (maximum and RMS) of the uncertainty of the orbital position of the Moon for 5 yr, assuming no observations since 2018, using only LLR data or LLR data with simulated VLBI data. Figure 3. Open in new tabDownload slide Estimation (maximum and RMS) of the uncertainty of the lunar physical libration (measured in maximum discrepancy across the lunar surface) for 5 yr, assuming no observations since 2018, using only LLR data or LLR data and simulated VLBI data. Figure 3. Open in new tabDownload slide Estimation (maximum and RMS) of the uncertainty of the lunar physical libration (measured in maximum discrepancy across the lunar surface) for 5 yr, assuming no observations since 2018, using only LLR data or LLR data and simulated VLBI data. 5.2 Lunar orientation parameters The formal errors of the daily ΔX and ΔZ detected from Earth–Moon VLBI simulations are shown in Fig. 4 (Legacy-1 subset of CONT17) and Fig. 5 (Legacy-2). It can be seen that both rotations are determined with uncertainty (3× the formal error) 0.3 mas or lower with Legacy-1. With the other subset, the uncertainties are ≈1.5× bigger, despite the larger number of observations than in the first subset. This comparison agrees with the original CONT17 estimations for the EOP determination. Figure 4. Open in new tabDownload slide Formal errors (1σ) of determined lunar orientation parameters (X and Z rotations) in the scenario of CONT17 Legacy-1 with lunar radio telescope. Figure 4. Open in new tabDownload slide Formal errors (1σ) of determined lunar orientation parameters (X and Z rotations) in the scenario of CONT17 Legacy-1 with lunar radio telescope. Figure 5. Open in new tabDownload slide Formal errors (1σ) of determined lunar orientation parameters (X and Z rotations) in the scenario of CONT17 Legacy-2 with lunar radio telescope. Figure 5. Open in new tabDownload slide Formal errors (1σ) of determined lunar orientation parameters (X and Z rotations) in the scenario of CONT17 Legacy-2 with lunar radio telescope. 5.3 Celestial reference frame In order to show how lunar VLBI observations improve the realization of the celestial reference frame, we estimate radio source positions using only CONT17 Earth-based observations and observations with Earth–Moon baselines. Results are shown in Figs 6 and 7 for right ascension and declination, respectively. The plots show formal errors of source positions for Earth and Earth–Moon observations. We can see that for a moderate number of sources one Moon-based telescope improves position accuracy by more than 10 times. Figure 6. Open in new tabDownload slide Formal errors (1σ) of determined right ascension of observed radio sources from CONT17 Legacy-1 observations, with and without Earth–Moon VLBI. Figure 6. Open in new tabDownload slide Formal errors (1σ) of determined right ascension of observed radio sources from CONT17 Legacy-1 observations, with and without Earth–Moon VLBI. Figure 7. Open in new tabDownload slide Formal errors (1σ) of determined declinations of observed radio sources from CONT17 Legacy-1 observations, with and without Earth–Moon VLBI. Figure 7. Open in new tabDownload slide Formal errors (1σ) of determined declinations of observed radio sources from CONT17 Legacy-1 observations, with and without Earth–Moon VLBI. The formal errors for the best sources of the International Celestial Reference Frame (ICRF) are about tens of mas (Fey et al. 2015), which is smaller than the Earth-only results in Figs 6 and 7. This is because our estimates are based on 9 d of data, while the ICRF is built using the results from 40 yr of VLBI observations. The Earth–Moon VLBI results, also estimated from 9 d of observations, will improve with a longer time-span. Fig. 8 shows the ratio of the formal errors of determined declinations without Earth–Moon VLBI to those with Earth–Moon VLBI, relative to ecliptic latitude. The ratio was multiplied by the square root of the number of observations with Earth–Moon VLBI divided by the number of observations without Earth–Moon VLBI. We can see that points shown in Fig. 8 are greater than 1 for all latitudes and more than 10 for about half of the sources. The improvements are the least for latitudes close to zero because the Earth–Moon baseline and source unit vectors are both close to the ecliptic plane. Figure 8. Open in new tabDownload slide Improvement factor (Earth and Earth–Moon VLBI versus just Earth VLBI) of radio source declination formal error normalized by the square root of the relative number of observations (as a function of ecliptic latitude). Figure 8. Open in new tabDownload slide Improvement factor (Earth and Earth–Moon VLBI versus just Earth VLBI) of radio source declination formal error normalized by the square root of the relative number of observations (as a function of ecliptic latitude). Interstellar scattering and intrinsic source structure effects can limit position errors obtained by Earth–Moon VLBI to greater than those shown in Figs 6 and 7. However, Earth–Moon VLBI can help us to estimate the source structure, in a manner similar to present VLBI observations (see Frey et al. 2018). There are techniques being developed to apply the source structure map to astrometric VLBI delay calculation. 5.4 General relativity tests The relativistic contribution by Solar system bodies to the VLBI time delay arises with baseline b. Using the equation from Klioner (1991), we computed the gravitational delay and delay due to translational motion for baselines b = 6000 km and |$b = 380\, 000$| km. Table 3 shows the delays for different baseline lengths (second column) and different angles between the source unit vector and baseline vector (third column). It is visible that delay generated by the Sun is detectable for two antennas located on Earth only if the angle is less than 30 degrees. Locating one of the antennas on the Moon will make that delay detectable on all angles. Table 3. Delay generated by gravity of a body (ΔtG) and by its translational motion (ΔtM). Body Baseline, km Angle ΔtG ΔtM Sun 6000 Grazing ray 169.24 ns 0.0082 ps Sun 6000 1 deg 45.33 ns Sun 6000 30 deg 1.47 ns Sun 6000 90 deg 0.40 ns Sun 6000 175 deg 17.25 ps Sun 380 000 Grazing ray 8588.24 ns 0.4166 ps Sun 380 000 1 deg 3098.72 ns Sun 380 000 30 deg 93.59 ns Sun 380 000 90 deg 25.02 ns Sun 380 000 175 deg 1.11 ns Body Baseline, km Angle ΔtG ΔtM Sun 6000 Grazing ray 169.24 ns 0.0082 ps Sun 6000 1 deg 45.33 ns Sun 6000 30 deg 1.47 ns Sun 6000 90 deg 0.40 ns Sun 6000 175 deg 17.25 ps Sun 380 000 Grazing ray 8588.24 ns 0.4166 ps Sun 380 000 1 deg 3098.72 ns Sun 380 000 30 deg 93.59 ns Sun 380 000 90 deg 25.02 ns Sun 380 000 175 deg 1.11 ns Open in new tab Table 3. Delay generated by gravity of a body (ΔtG) and by its translational motion (ΔtM). Body Baseline, km Angle ΔtG ΔtM Sun 6000 Grazing ray 169.24 ns 0.0082 ps Sun 6000 1 deg 45.33 ns Sun 6000 30 deg 1.47 ns Sun 6000 90 deg 0.40 ns Sun 6000 175 deg 17.25 ps Sun 380 000 Grazing ray 8588.24 ns 0.4166 ps Sun 380 000 1 deg 3098.72 ns Sun 380 000 30 deg 93.59 ns Sun 380 000 90 deg 25.02 ns Sun 380 000 175 deg 1.11 ns Body Baseline, km Angle ΔtG ΔtM Sun 6000 Grazing ray 169.24 ns 0.0082 ps Sun 6000 1 deg 45.33 ns Sun 6000 30 deg 1.47 ns Sun 6000 90 deg 0.40 ns Sun 6000 175 deg 17.25 ps Sun 380 000 Grazing ray 8588.24 ns 0.4166 ps Sun 380 000 1 deg 3098.72 ns Sun 380 000 30 deg 93.59 ns Sun 380 000 90 deg 25.02 ns Sun 380 000 175 deg 1.11 ns Open in new tab The model of VLBI time delay contains the PPN parameter γ, which characterizes space curvature produced by unit rest mass. The accuracy of γ estimated by the initial CONT17 Legacy-1 VLBI series 2.0 × 10−3 has been obtained, while the estimation from Legacy-1 and Legacy-2 gives 1.3 × 10−3. Adding Earth–Moon VLBI improves the accuracy to 0.5 × 10−3 for Legacy 1 and to 0.06 × 10−3 for Legacy-1 and Legacy-2. The experiment was not specially designed for estimation of γ and does not have sources close to the Sun or Jupiter. Nevertheless, we obtain an improvement of a factor from 4 to 21. A dedicated experiment like that of Titov et al. (2018) with Earth–Moon VLBI can overcome the present best accuracy of 0.23 × 10−4 obtained from frequency shift measurements of theCassini spacecraft (Bertotti, Iess & Tortora 2003). Epstein & Shapiro (1980) also show that post-post-Newtonian deflection of the Sun is about 11 μarcsec, which corresponds to a delay equal to 1.1 ps for b = 6000 km and 67.6 ps for |$b = 380\, 000$| km. Therefore, growth of the baseline will increase the accuracy of the detection of relativistic effects. 6 CIRCUMLUNAR ORBIT DETERMINATION SIMULATION A dedicated numerical simulation was performed to study how an improvement in the lunar reference frame (Table 1) affects the determination of orbits of an example lunar satellite constellation. Nine satellites were modelled with circular inclined orbits with a semimajor axis of 4500 km. The model observations were inter-satellite link (ISL) measurements and two-way range measurements to the retroreflector points. Table 4 shows the accuracy of the determination of orbits of the given constellation projected to the average lunar surface point line of sight for two scenarios of reference point uncertainties according to Table 1. ISL measurement accuracies were assumed to be at the level of a few centimetres (1σ) in order to make the results sensitive to the differences between the reference point uncertainties. A solution is produced with the least-squares method using the measurement interval of 3.5 h equal to about half of the orbital period and the following propagation interval of 12 h. Statistics were taken over all satellites in the constellation. Three values are provided for each time interval: RMS value, maximum value over 95 per cent confidence interval (CI) and maximum value overall. Table 4. Range accuracy of orbit determination example based on ISL and two-way range measurements to the reference points with different uncertainties. Reference points by: LLR only LLR and VLBI RMS (3.5 h measurements) 7.9 cm 2.7 cm 95% CI (3.5 h measurements) 12.9 cm 4.8 cm Max. (3.5 h measurements) 16.2 cm 8.0 cm RMS (12 h propagation) 14.8 cm 7.8 cm 95% CI (12 h propagation) 26.5 cm 14.5 cm Max. (12 h propagation) 43.3 cm 24.4 cm Reference points by: LLR only LLR and VLBI RMS (3.5 h measurements) 7.9 cm 2.7 cm 95% CI (3.5 h measurements) 12.9 cm 4.8 cm Max. (3.5 h measurements) 16.2 cm 8.0 cm RMS (12 h propagation) 14.8 cm 7.8 cm 95% CI (12 h propagation) 26.5 cm 14.5 cm Max. (12 h propagation) 43.3 cm 24.4 cm Open in new tab Table 4. Range accuracy of orbit determination example based on ISL and two-way range measurements to the reference points with different uncertainties. Reference points by: LLR only LLR and VLBI RMS (3.5 h measurements) 7.9 cm 2.7 cm 95% CI (3.5 h measurements) 12.9 cm 4.8 cm Max. (3.5 h measurements) 16.2 cm 8.0 cm RMS (12 h propagation) 14.8 cm 7.8 cm 95% CI (12 h propagation) 26.5 cm 14.5 cm Max. (12 h propagation) 43.3 cm 24.4 cm Reference points by: LLR only LLR and VLBI RMS (3.5 h measurements) 7.9 cm 2.7 cm 95% CI (3.5 h measurements) 12.9 cm 4.8 cm Max. (3.5 h measurements) 16.2 cm 8.0 cm RMS (12 h propagation) 14.8 cm 7.8 cm 95% CI (12 h propagation) 26.5 cm 14.5 cm Max. (12 h propagation) 43.3 cm 24.4 cm Open in new tab The results show a ≈ 2× improvement in orbit determination accuracy after the suggested Earth–Moon VLBI implementation. 7 CONCLUSION For the purposes of astrometry, the placement of the lunar radio telescope near the equator, close to the edge of the visible side of the lunar disc, is preferred. The obtained results are based on the assumption that the sensitivity of the proposed Earth–Moon interferometer will be enough for obtaining an accurate time delay (i.e. the existence of sufficiently bright and compact quasar structures is assumed). Just 9 d of intensive Earth–Moon VLBI observations will improve the accuracy of the Earth–Moon reference frame by 3–6×, and the accuracy of the lunar reference frame by 4–10×. The accuracy of the lunar ephemeris (both orbit and physical libration), presently calculated from 47 yr of LRR, will improve from metres to decimetres for a 5-yr interval into the future. The accuracy of daily corrections to lunar orientation reach 0.3 mas (≈2.5 mm on the lunar surface) for X and Z rotations. With the chosen location of the instrument, Y rotations cannot be determined from Earth–Moon VLBI; however, LLR is capable of determining these, with an accuracy of 3–5 mas (centimetres on the lunar surface). According to simulated observations for 9 d, the accuracy of the celestial reference frame can be improved more than 10× for about half of the radio sources distant from the ecliptic. The Earth–Moon interferometer will have higher sensitivity to the relativistic parameters of post-Newtonian and post-post-Newtonian formalism and can be used for relativistic tests. The lunar reference frame, improved by Earth–Moon VLBI, will bring a ≈2× improvement in accuracy of the determination of orbits of a possible lunar satellite constellation. However, such an improvement, while important for fundamental lunar science, will not be critical for applications such as lunar navigation with the present requirements. Some of the results promised by Earth–Moon VLBI relevant to study of the Moon itself could, in principle, be accomplished with other tools (more precise LLR, lunar radio transponders, lunar optical telescope), while orbital radio telescopes are useful for astrophysics. However, Earth–Moon VLBI will stand unrivalled for future radio astrometry and building of the Earth–Moon reference frame. ACKNOWLEDGEMENTS The authors from the Institute of Applied Astronomy of the Russian Academy of Sciences are grateful to Iskander Gayazov, Alexander Ipatov, Dmitry Vavilov and numerous other colleagues for helpful discussions, advice and support throughout this work. 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TI - Earth–Moon very-long-baseline interferometry project: modelling of the scientific outcome
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/stz827
DA - 2019-06-11
UR - https://www.deepdyve.com/lp/oxford-university-press/earth-moon-very-long-baseline-interferometry-project-modelling-of-the-7ZEbPKrHrg
SP - 815
VL - 486
IS - 1
DP - DeepDyve
ER -