The simulation of lunar gravity field recovery from D-VLBI of Chang’E-1 and SELENE lunar orbiters

The lunar gravity field is a foundation to study the lunar interior structure, and to recover the evolution history of the Moon. It is still an open and key topic for lunar science. For above mentioned reasons, it becomes one of the important scientific objectives of recent lunar missions, such as KAGUYA (SELENE) the Japanese lunar mission and Chang’E-1, the Chinese lunar mission. The Chang’E-1 and the SELENE were successfully launched in 2007. It is estimated that these two missions can fly around the Moon longer than 6 months simultaneously. In these two missions, the Chinese new VLBI (Very Long Baseline Interferometry) network will be applied for precise orbit determination (POD) by using a differential VLBI (D-VLBI) method during the mission period. The same-beam D-VLBI technique will contribute to recover the lunar gravity field together with other conventional observables, i.e. R&RR (Range and Range Rate) and multi-way Doppler. Taking VLBI tracking conditions into consideration and using the GEODYNII/SOVLE software of GSFC/NASA/USA [8 and 10], we simulated the lunar gravity field recovering ability with and without D-VLBI between the Chang’E-1 and SELENE main satellite. The cases of overlapped flying and tracking period of 30 days, 60 days and 90 days have been analyzed, respectively. The results show that D-VLBI tracking between two lunar satellites can improve the gravity field recovery remarkably. The results and methods introduced in this paper will benefit the actual missions.     

Chang’E-1 precision orbit determination and lunar gravity field solution

In this paper we present results assessing the role of Very Long Baseline Interferometry (VLBI) tracking data through precision orbit determination (POD) during the check-out phase for Chang’E-1, and the lunar gravity field solution CEGM-01 based on the orbital tracking data acquired during the nominal phase of the mission. The POD of Chang’E-1 is performed using S-band two-way Range and Range Rate (R&RR) data, together with VLBI delay and delay rate observations. The role of the VLBI data in the POD of Chang’E-1 is analyzed, and the resulting orbital accuracies are estimated for different solution strategies. The final orbital accuracies proved that the VLBI tracking data can improve the Chang’E-1 POD significantly. Consequently, CEGM-01 based on six-month tracking data during Chang’E-1 nominal mission phase is presented, and the accuracy of the model is assessed by means of the gravity field power spectrum, admittance and coherence between gravity and topography, lunar surface gravity anomaly and POD for both Chang’E-1 and Lunar Prospector (LP). Our analysis indicates that CEGM-01 has significant improvements over a prior model (i.e. GLGM-2), and shows the potential of Chang’E-1 tracking data in high resolution lunar gravity field model solution by combining with SELENE and LP tracking data.     

Long-wavelength lunar gravity field recovery from simulated orbit and inter-satellite tracking data

As has been demonstrated recently, inter-satellite Ka-band tracking data collected by the GRAIL (Gravity Recovery And Interior Laboratory) spacecraft have the potential to improve the resolution and accuracy of the lunar gravity field by several orders of magnitude compared to previous models. By means of a series of simulation studies, here we investigate the contribution of inter-satellite ranging for the recovery of the Moon’s gravitational features; the evaluation of results is made against findings from ground-based Doppler tracking. For this purpose we make use of classical dynamic orbit determination, supported by the analysis of satellite-to-satellite tracking observations. This study sheds particularly light on the influence of the angular distance between the two satellites, solar radiation modeling and the co-estimation of the lunar Love number k₂. The quality of the obtained results is assessed by gravity field power spectra, gravity anomalies and precision orbit determination. We expect our simulation results to be supportive for the processing of real GRAIL data.     

Simulation study for the determination of the lunar gravity field from PRARE-L tracking onboard the German LEO mission

A simulation study has been performed at GFZ Potsdam, which shows the anticipated improvement of the lunar gravity field model with respect to current (LP150Q model) or near-future (SELENE) knowledge in the framework of the planned German Lunar Explorations Orbiter (LEO) mission, based on PRARE-L (Precise Range And Range-rate Equipment – Lunar version) Satellite-to-Satellite (SST) and Satellite-Earth-Satellite (SEST) tracking observations. It is shown that the global mean error of the lunar gravity field can be reduced to less than 0.1 mGal at a spatial resolution of 50 km. In the spectral domain, this means a factor of 10 (long wavelengths) and some 100 (mid to short wavelengths) improvement as compared to predictions for SELENE or a factor of 1000 with respect to LP150Q. Furthermore, a higher spatial resolution of up to 28 km seems feasible and would correspond to a factor of 2–3 improvement of SELENE results. Moreover, PRARE-L is expected to derive the low-degree coefficients of the lunar gravity field with unprecedented accuracy. Considering long mission duration (at least 1 year is planned) this would allow for the first time a precise direct determination of the low-degree tidal Love numbers of the Moon and, in combination with high precision SEST, would provide an experimental basis to study relativistic effects such as the periselenium advance in the Earth–Moon system.     

A simulation study for anticipated accuracy of lunar gravity field model by SELENE tracking data

Results of numerical simulations are presented to examine the global gravity field recovery capability of the Japanese lunar exploration project SELENE (SELenological and ENgineering Explorer) which will be launched in 2007. New characteristics of the SELENE lunar gravimetry include 4-way satellite-to-satellite Doppler tracking of main orbiter and differential VLBI tracking of two small free-flier satellites. It is shown that the proposed satellite constellation will provide the first truly global satellite tracking data coverage. The expected results from these data are; (1) drastic reduction in far-side gravity error, (2) estimation of many gravity coefficients by the observation, not by a priori information, and (3) one order of magnitude improvement over existing gravity models for low-degree field.     

Impact of covariance information of kinematic positions on orbit reconstruction and gravity field recovery

Gravity missions are equipped with onboard Global Positioning System (GPS) receivers for precise orbit determination (POD) and for the extraction of the long wavelength part of the Earth’s gravity field. As positions of low Earth orbiters (LEOs) may be determined from GPS measurements at each observation epoch by geometric means only, it is attractive to derive such kinematic positions in a first step and to use them in a second step as pseudo-observations for gravity field determination. The drawback of not directly using the original GPS measurements is, however, that kinematic positions are correlated due to the ambiguities in the GPS carrier phase observations, which in principle requires covariance information be taken into account. We use GRACE data to show that dynamic or reduced-dynamic orbit parameters are not optimally reconstructed from kinematic positions when only taking epoch-wise covariance information into account, but that essentially the same orbit quality can be achieved as when directly using the GPS measurements, if correlations in time are taken into account over sufficiently long intervals. For orbit reconstruction covariances have to be considered up to one revolution period to avoid ambiguity-induced variations of kinematic positions being erroneously interpreted as orbital variations. For gravity field recovery the advantage is, however, not very pronounced.     

Proposal of application of same beam VLBI measurements in precision orbit determination of lunar orbiter and return capsule and lunar gravity field simulation in Chang’E-3 mission

High accuracy differenced phase delay can be obtained by observing multiple point frequencies of two spacecraft using the same beam Very Long Baseline Interferometry (VLBI) technology. Its contribution in lunar spacecraft precision orbit determination has been performed during the Japanese lunar exploration mission SELENE. In consideration that there will be an orbiter and a return capsule flying around the moon during the Chinese lunar exploration future mission Chang’E-3, the contributions of the same beam VLBI in spacecraft precision orbit determination and lunar gravity field solution have been investigated. Our results show that the accuracy of precision orbit determination can be improved more than one order of magnitude after including the same beam VLBI measurements. There are significant improvements in accuracy of low and medium degree coefficients of lunar gravity field model obtained from combination of two way range and Doppler and the same beam VLBI measurements than the one that only uses two way range and Doppler data, and the accuracy of precision orbit determination can reach meter level.     

Comparison analyses on the 150 × 150 lunar gravity field models by gravity/topography admittance,correlation and precision orbit determination

We analyzed the 150 × 150 lunar gravity field models, LP150Q, GLGM-3 and SGM150, using the power spectrum on the lunar nearside and farside, the lunar global and localized gravity/topography admittance and correlation, and Chang’E-2 precision orbit determination to investigate which model is a more effective tool to estimate geophysical parameters and determine the lunar satellite precision orbit. Results indicate that all gravity field models can be used to estimate the lunar geophysical parameters of the nearside of the Moon. However, SGM150 is better in such computation of the farside. Additionally, SGM150 is shown to be the most useful model for determining the lunar satellite orbit.     

GRACE gravity field modeling with an investigation on correlation between nuisance parameters and gravity field coefficients

The GRACE (Gravity Recovery And Climate Experiment) monthly gravity models have been independently produced and published by several research institutions, such as Center for Space Research (CSR), GeoForschungsZentrum (GFZ), Jet Propulsion Laboratory (JPL), Centre National d’Etudes Spatiales (CNES) and Delft Institute of Earth Observation and Space Systems (DEOS). According to their processing standards, above institutions use the traditional variational approach except that the DEOS exploits the acceleration approach. The background force models employed are rather similar. The produced gravity field models generally agree with one another in the spatial pattern. However, there are some discrepancies in the gravity signal amplitude between solutions produced by different institutions. In particular, 10%–30% signal amplitude differences in some river basins can be observed. In this paper, we implemented a variant of the traditional variational approach and computed two sets of monthly gravity field solutions using the data from January 2005 to December 2006. The input data are K-band range-rates (KBRR) and kinematic orbits of GRACE satellites. The main difference in the production of our two types of models is how to deal with nuisance parameters. This type of parameters is necessary to absorb low-frequency errors in the data, which are mainly the aliasing and instrument errors. One way is to remove the nuisance parameters before estimating the geopotential coefficients, called NPARB approach in the paper. The other way is to estimate the nuisance parameters and geopotential coefficients simultaneously, called NPESS approach. These two types of solutions mainly differ in geopotential coefficients from degree 2 to 5. This can be explained by the fact that the nuisance parameters and the gravity field coefficients are highly correlated, particularly at low degrees. We compare these solutions with the official and published ones by means of spectral analysis. It is found that our solutions are, in general, consistent with others in the spatial pattern. The water storage variations of the Amazon, Chari and Ganges river basins have also been computed. The variations computed with the NPARB approach are closer to those produced by JPL and DEOS solutions, while the variations produced with the NPESS approach are in good agreement with those produced by the CSR and GFZ solutions. A simulation study is implemented with considering realistic noise and low-frequency error. The two approaches are used to recover the true model. The NPESS solution appears closer to the true one. Therefore we are inclined to estimate the nuisance parameters simultaneously with the geopential coefficients.     

Lunar gravity field modeling critical analysis and challenges

This paper summarizes and provides a critical analysis of the historical developments of lunar gravitational models from the earliest use of ground based tracking systems of the Lunar Orbiter to the Lunar Prospector mission. This encompasses a comprehensive and critical analysis of the various methods used in the estimation of the gravity coefficients and the processing of large batches of diverse measurements and data types. It has been shown that weakness exists in the current models of the lunar gravity field, which is primarily due to the lack of far side lunar tracking data information, which makes the lunar potential modeling difficult but expected to be overcome as data from SELENE satellite-to-satellite tracking becomes available. Comparisons of various lunar models reveal an agreement in the low order coefficients of the spherical harmonics. However, substantial differences in the models exist in the higher-order harmonics. A numerical comparison has been presented showing the performance of all the contemporary lunar gravitational models used within the astrodynamics community and available in public domain. Improvements to the current models are part of a continuing process and the recent model improvements and future possibilities in lunar gravity modeling are discussed. A brief review of the recent missions has been presented. It is hoped that this critical review will benefit the researchers by presenting the historical as well as state of the art in this field.     

On the usefulness of atmospheric and oceanic angular momentum in recovering polar motion and gravity field variations in a unified process

Earth rotation parameters (ERPs) are excited by variations in the mass distribution on the Earth’s surface and the exchange of angular momentum between the atmosphere and oceans and the solid Earth. The same mass redistribution causes temporal changes in the gravity field coefficients with the second degree harmonics related to the rotational deformation and hence to changes in the Earth’s inertial tensor. If precise models of the atmospheric and oceanic angular momentum (AM) are available solution for polar motion and degree 2 Stokes harmonics can be unified. In this study we utilize SLR tracking of LAGEOS to compare (i) degree 2 harmonics from ERPs and gravitation, and (ii) LAGEOS excitation functions and geophysical data (mass + motion). Finally, we investigate to what extent a unified approach is possible with current models for AM data and gravity mass change estimated from ERP within orbit determinations.     

European present and future scientific activities in the field of low gravity research

This report deals with the present plans and philosophy of the European Space Agency as regards the interest and the justification for a European Experimental Research Program in Space. It offers recommendations to be considered by both the space investigators and the E.S.A. executive in order to achieve the most urgent needs.     

Integral inversion of GRAIL inter-satellite gravitational accelerations for regional recovery of the lunar gravitational field

We present an integral-based approach for high-resolution regional recovery of the gravitational field in this article. We derive rigorous remove-compute-restore integral estimators relating the line-of-sight gravitational acceleration to an arbitrary order radial derivative of the gravitational potential. The integral estimators are composed of three terms, i.e., the truncated integration, the low-frequency line-of-sight gravitational acceleration, and the high-frequency truncation error (effect of the distant zones). We test the accuracy of the integral transformations and of the integral estimators in a closed-loop simulation over the Montes Jura region on the nearside of the Moon. In this way, we determine optimal sizes of integration radii and grid discretisation. In addition, we investigate the performance of the regional integral inversion with synthetic and realistic GRAIL observations. We demonstrate that the regional inversion results of the disturbing gravitational potential and its first order radial derivative in the Montes Jura mountain range are less contaminated by high-frequency noise than the global spherical harmonic models.     

GPS-only gravity field recovery with GOCE,CHAMP, and GRACE

Gravity missions such as the Gravity field and steady-state Ocean Circulation Explorer (GOCE) are equipped with onboard Global Positioning System (GPS) receivers for precise orbit determination (POD), instrument time-tagging, and the extraction of the long wavelength part of the Earth’s gravity field. The very low orbital altitude of the GOCE satellite and the availability of dense 1 s GPS tracking data are ideal characteristics to exploit the contribution of GPS high-low Satellite-to-Satellite Tracking (hl-SST) to gravity field determination. We present gravity field solutions based on about 8 months of GOCE GPS hl-SST data from 2009 and compare the results with those obtained from the CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) missions. The very low orbital altitude of GOCE significantly improves gravity field recovery from GPS hl-SST data above degree 20, but not for the degrees below 20, where the quality of the spherical harmonic coefficients remains essentially unchanged. Despite the limited time span of GOCE data used, the gravity field of the Earth can be resolved up to about degree 115 using GPS data only. Empirically determined phase center variations (PCVs) of the GOCE onboard GPS helix antenna are, however, mandatory to achieve this performance.     

An alternative computation of a gravity field model from GOCE

GOCE is the first satellite with a gravitational gradiometer (SGG). This allows to determine a gravity field model with high spatial resolution and high accuracy. Four of the six independent components of the gravitational gradient tensors (GGT) are measured with high accuracy in the so-called measurement band (MB) from 5 to 100 mHz by the GOCE gradiometer. Based on more than 1 year of GOCE measurements, two gravity field models have been derived. Here, we introduce a strategy for spherical harmonic analysis (SHA) from GOCE measurements, with a bandpass filter applied to the SGG data, combined with orbit analysis based on the integral equation approach, and additional constraints (or stabilization) in the polar areas where no observation is available due to the orbit geometry. In addition, we combined the GOCE SGG part with a set of GRACE normal equations. This improves the accuracy of the gravity field in the long-wavelength parts, due to the complementarity of GOCE and GRACE. Comparison with other models and with external data shows that our results are rather close to the GPS-levelling data in well-selected test regions, with an uncertainty of 4–7 cm, for truncation at degree 200.     

Movement of a material point in the gravity field of a homogeneous ring

A ring-like mass exerts a well computable gravitational attraction on a material point located along a straight line, being perpendicular to the plane of the ring in its centre. In the state of weightlessness, an oscillatory movement will develop owing to this effect. The period, T, of oscillation depends on the gravity constant, on the density and dimensions of the ring, as well as on the amplitude of the oscillation. Its exact computation can be based on the determination of the gravity potential function of the ring. The oscillation has the following form:

$\text{T=}\text{1}\text{√f√}\textcolor{red}{}\text{.I(r,R,z)}$

where f is the gravitational constant.

is the density of the ring, r and R are the radii of the ring, z is the distance of the turning point of the oscillation from the centre of the ring, while I/r,R,z/ is an improper integral which can be computed with any desired accuracy owing to the favourable function-theoretical character of the potential. We computed the oscillation period for various possible values of the parameters and obtained time data of an order of magnitude which falls into a well observable interval.The outlined conceptual experiment for the improved determination of the gravitational constant may present, of course, many technical difficulties and error sources /e.g. the path of the oscillating point is quite unstable owing to the extremely small acting forces, electric charges and also radiation pressure might be present, the gravity field would show a gradient on the spot of the experiment, etc./. Nevertheless, it seems to be worthwhile to consider carrying out such an experiment, using the possibilities offered by modern techniques in observing distances and time. For the path distortions caused by errors, we give a few estimates, but in case of realization of the experiment, a more detailed error analysis must be made.     

Contribution of satellite laser ranging to combined gravity field models

In the framework of satellite-only gravity field modeling, satellite laser ranging (SLR) data is typically exploited to recover long-wavelength features. This contribution provides a detailed discussion of the SLR component of GOCO02S, the latest release of combined models within the GOCO series. Over a period of five years (January 2006 to December 2010), observations to LAGEOS-1, LAGEOS-2, Ajisai, Stella, and Starlette were analyzed. We conducted a series of closed-loop simulations and found that estimating monthly sets of spherical harmonic coefficients beyond degree five leads to exceedingly ill-posed normal equation systems. Therefore, we adopted degree five as the spectral resolution for real data analysis. We compared our monthly coefficient estimates of degree two with SLR and Gravity Recovery and Climate Experiment (GRACE) time series provided by the Center for Space Research (CSR) at Austin, Texas. Significant deviations in C₂₀ were noted between SLR and GRACE; the agreement is better for the non-zonal coefficients. Fitting sinusoids together with a linear trend to our C₂₀ time series yielded a rate of (−1.75 ± 0.6) × 10⁻¹¹/yr; this drift is equivalent to a geoid change from pole to equator of 0.35 ± 0.12 mm/yr or an apparent Greenland mass loss of 178.5 ± 61.2 km³/yr. The mean of all monthly solutions, averaged over the five-year period, served as input for the satellite-only model GOCO02S. The contribution of SLR to the combined gravity field model is highest for C₂₀, and hence is essential for the determination of the Earth’s oblateness.     

Integration of In-Situ Resource Utilization into lunar/Mars exploration through field analogs

The ability to extract and process resources at the site of exploration into useful products such as propellants, life support and power system consumables, and radiation and rocket exhaust plume debris shielding, known as In-Situ Resource Utilization or ISRU, has the potential to significantly reduce the launch mass, risk, and cost of robotic and human exploration of space. The incorporation of ISRU into missions can also significantly influence technology selection and system development in other areas such as power, life support, and propulsion. For example, the ability to extract or produce large amounts of oxygen and/or water in-situ could minimize the need to completely close life support air and water processing system cycles, change thermal and radiation protection of habitats, and influence propellant selection for ascent vehicles and surface propulsive hoppers. While concepts and even laboratory work on evaluating and developing ISRU techniques such as oxygen extraction from lunar regolith have been going on since before the Apollo 11 Moon landing, no ISRU system has ever flown in space, and only recently have ISRU technologies been developed at a scale and at a system level that is relevant to actual robotic and human mission applications. Because ISRU hardware and systems have never been demonstrated or utilized before on robotic or human missions, architecture and mission planners and surface system hardware developers are hesitant to rely on ISRU products and services that are critical to mission and system implementation success. To build confidence in ISRU systems for future missions and assess how ISRU systems can best influence and integrate with other surface system elements, NASA, with international partners, are performing analog field tests to understand how to take advantage of ISRU capabilities and benefits with the minimum of risk associated with introducing this game-changing approach to exploration. This paper will describe and review the results of four analog field tests (Moses Lake in 6/08, Mauna Kea in 11/08, Flagstaff in 9/09, and Mauna Kea in 1/10) that have begun the process of integrating ISRU into robotic and human exploration systems and missions, and propose future ISRU-related analog field test activities that can be performed in collaboration with non-US space agencies.