GOCE: The Earth Gravity Field by Space Gradiometry

An artificial satellite, flying in a purely gravitational field is a natural probe, such that, by a very accurate orbit determination, would allow a perfect estimation of the field. A true satellite experiences a number of perturbational, non-gravitational forces acting on the shell of the spacecraft; these can be revealed and accurately measured by a spaceborne accelerometer. If more accelerometers are flown in the same satellite, they naturally eliminate (to some extent) the common perturbational accelerations and their differences are affected by the second derivatives of the gravity fields only (gradiometry). The mission GOCE is based on this principle. Its peculiar dynamical observation equations are reviewed. The possibility of estimating the gravity field up to some harmonic degree (200) is illustrated.     

Future lunar gravity measurements

Future missions to the Moon should include a detailed high-resolution global gravity survey from a low (15–30 km) polar orbiting spacecraft. The use of gravity gradiometer instruments on board the spacecraft will give higher-resolution data at lower total mission cost that the present Doppler tracking technique. Simulations show that although a three axis gradiometer system is preferred, and can even be used to estimate spacecraft attitude and altitude variation, a properly oriented single rotating gravity gradiometer can be used to resolve closely spaced mascons in both the along-track and cross-track directions.Paper presented at theFuture Lunar Exploration session of the Tenth Lunar and Planetary Science. Conference, Johnson Space Center, Houston, Texas, 19–23 March 1979.     

Future Satellite Gravimetry for Geodesy

After GRACE and GOCE there will still be need and room for improvement of the knowledge (1) of the static gravity field at spatial scales between 40 km and 100 km, and (2) of the time varying gravity field at scales smaller than 500 km. This is shown based on the analysis of spectral signal power of various gravity field components and on the comparison with current knowledge and expected performance of GRACE and GOCE. Both, accuracy and resolution can be improved by future dedicated gravity satellite missions. For applications in geodesy, the spectral omission error due to the limited spatial resolution of a gravity satellite mission is a limiting factor. The recommended strategy is to extend as far as possible the spatial resolution of future missions, and to improve at the same time the modelling of the very small scale components using terrestrial gravity information and topographic models.We discuss the geodetic needs in improved gravity models in the areas of precise height systems, GNSS levelling, inertial navigation and precise orbit determination. Today global height systems with a 1 cm accuracy are required for sea level and ocean circulation studies. This can be achieved by a future satellite mission with higher spatial resolution in combination with improved local and regional gravity field modelling. A similar strategy could improve the very economic method of determination of physical heights by GNSS levelling from the decimeter to the centimeter level. In inertial vehicle navigation, in particular in sub-marine, aircraft and missile guidance, any improvement of global gravity field models would help to improve reliability and the radius of operation.     

Gravity gradient mapping from the lunar polar orbiter - a simulation study

Simulations of the gravity data to be expected from a Lunar Polar Orbiter spacecraft utilizing either a Doppler velocity tracking system or a gravity gradiometer instrument system are generated using a point mass model that gives an excellent representation of the types of gravity anomalies to be found on the Moon. If the state of the art in instrumentation of both systems remain at the level of ±1 mm/sec at 10 sec integration time for the Doppler velocity system accuracy and at ±1 Eotvos at 10 sec integration time for the gravity gradiometer system accuracy, inspection of the simulations indicates that a gravity gradiometer system will give science data with better resolution and higher amplitude-to-measurement noise ratio than the Doppler velocity system at altitudes below 100 km. The error model used in the study is one where the system errors are assumed to be dominated by the point measurement noise and data quantization noise. The effects of other, more controllable, systematic error sources are not considered in this simplified analysis. For example, both systems will be affected by errors in LPO orbital altitude and position knowledge, spacecraft maneuvers, and data reduction errors. In addition, a Doppler tracking system will be sensitive to errors produced by spacecraft acceleration (from outgassing or solar pressure) and poor relative position of the LPO, Relay Satellite and ground tracking station, while a gravity gradiometer system will be sensitive to errors from spacecraft attitude and angular rates. These preliminary study results now need to be verified by a more complete error analysis in which all the uncertainties of the data gathering process are formally mapped into uncertainties in the resulting gravity maps.     

Efficient Determination of Global Gravity Field from Satellite-to-satellite Tracking Mission

The gravity field dedicated satellite missions like CHAMP, GRACE, and GOCE are supposed to map the Earth's global gravity field with unprecedented accuracy and resolution. New models of the Earth's static and time-variable gravity fields will be available every month as one of the science products from GRACE. A method for the efficient gravity field recovery is presented using in situ satellite-to-satellite observations at altitude and results on static as well as temporal gravity field recovery are shown. Considering the energy relationship between the kinetic energy of the satellite and the gravitational potential, the disturbing potential observations can be computed from the orbital state vector, using high-low GPS tracking data, low–low satellite-to-satellite GRACE measurements, and data from 3-axis accelerometers. The solution method is based on the conjugate gradient iterative approach to efficiently recover the gravity field coefficients and approximate error covariance up to degree and order 120 every month. Based on the monthly GRACE noise-only simulation, the geoid was obtained with an accuracy of a few cm and with a resolution (half wavelength) of 160 km. However, the geoid accuracy can become worse by a factor of 6–7 because of spatial aliasing. The approximate error covariance was found to be a very good accuracy measure of the estimated coefficients, geoid, and gravity anomaly. The temporal gravity field, representing the monthly mean continental water mass redistribution, was recovered in the presence of measurement noise and high frequency temporal variation. The resulting recovered temporal gravity fields have about 0.3 mm errors in terms of geoid height with a resolution of 670 km.     

Challenges From Solid Earth Dynamics for Satellite Gravity Field Missions in the Post-Goce Era

Examples from four main categories of solid-earth deformation processes are discussed for which the GOCE and GRACE satellite gravity missions will not provide a high enough spatial or temporal resolution or a sufficient accuracy. Quasi-static and episodic solid-earth deformation would benefit from a new satellite gravity mission that would provide a higher combined spatial and temporal resolution. Seismic and core periodic motions would benefit from a new satellite mission that would be able to detect gravity variations with a higher temporal resolution combined with very high accuracies.     

A Proposed lunar orbiting gravity gradiometer experiment

Analysis of the gravity gradiometer developed by R. L. Forward and C. C. Bell at the Hughes Research Laboratories suggest than an accuracy, in the range 0.1 to 0.5 EU can be expected in a lunar orbiter application. This accuracy will allow gradient anomalies associated with mascons to be mapped with 1% accuracy and should reveal a great deal of new information about the lunar gravity field.The proposed experiment calls for putting such a gradiometer into a closely circular polar orbit at an average height of about 30 km above the lunar surface. This orbit allows the entire lunar surface to be covered in fourteen days, the gradiometer to be checked twice per revolution and results in successive passes above the lunar surface being spaced at about the resolution limit of about 30 km set both by the satellite altitude and instrumental integration time. Doppler tracking will be employed and the spacecraft will carry an electromagnetic altimeter. Gradient and altitude data from the far side of the Moon can be stored for replay when communication is re-established.     

日本SELENE月球探测计划和卫星间多普勒跟踪的数学模型

日本月球探测计划（SELENE）定于2004年夏季利用HIIa火箭发射一组共3颗绕月人造卫星。他们是主卫星、跟踪中断卫星和空间VLBI电波源。其主要科学目标之一是利用对绕月卫星的多普勒跟踪数据精确测定月球重力场,研究月球的起源与演化。SELENE计划中实现这个科学目标的关键技术是引入中继卫星,目的在于当处于低轨道的主卫星飞行到月球背面地面观测站无法观测时,采用卫星间跟踪方法（SST）,建立地面站与主卫星之间的联系,以得到月球背面重力场的直接测量数据。介绍了几种典型的四程卫星间多普勒跟踪模式和相应的数学模型,并针对SELENE计划中采用的特殊四程多普勒跟踪模式建立了卫星相对观测站速度与跟踪信号多普勒频移之间的转换关系。提出了利用GEODYNⅡ定轨分析软件处理SELENE多普勒跟踪数据的流程。     

Gravity field and rotation state of Mercury from the BepiColombo Radio Science Experiments

The ESA mission BepiColombo will include a Mercury Planetary Orbiter equipped with a full complement of instruments to perform Radio Science Experiments. Very precise range and range-rate tracking from Earth, on-board accelerometry, altimetry and accurate angular measurements with optical instruments will provide large data sets. From these it will be possible to study (1) the global gravity field of Mercury and its temporal variations due to tides, (2) the medium to short scale (down do 300400 km) gravity anomalies, (3) the rotation state of the planet, in particular the obliquity and the libration with respect to the 3/2 spin orbit resonance and (4) the orbit of the center of mass of the planet.With the global gravity field and the rotation state it is possible to tightly constrain the internal structure of the planet, in particular to determine whether the solid surface of the planet is decoupled from the inner core by some liquid layer, as postulated by dynamo theories of Mercury's magnetic field. With the gravity anomalies and altimetry it is possible to study the geophysics of the planet's crust, mantle and impact basins. With the orbit of the planet closest to the Sun it is possible to constrain relativistic theories of gravitation.The possibility of achieving these scientific goals has been tested with a full cycle numerical simulation of the Radio Science Experiments. It includes the generation of simulated tracking and accelerometer data, and the determination, by least squares fit, of a long list of variables including the initial conditions for each observed arc, calibration parameters, gravity field harmonic coefficients, and corrections to the orbit of Mercury. An error budget has been deduced both from the formal covariance matrices and from the actual difference between the nominal values used in the data simulation and the solution. Thus the most complete error budget contains the effect of systematic measurement errors and is by far more reliable than a formal one. For the rotation experiment an error budget has been computed on the basis of dedicated studies on each separate error source.The results of the full cycle simulation are positive, that is the experiments are feasible at the required level of accuracy. However, the extraction of the full accuracy results from the data will be by no means trivial, and there are a number of open problems, both in the data processing (e.g., the selection of the orbital arc length) and in the mission scheduling (e.g., the selection of the target areas for the rotation experiment).     

ISA accelerometer onboard the Mercury Planetary Orbiter: error budget

We have estimated a preliminary error budget for the Italian Spring Accelerometer (ISA) that will be allocated onboard the Mercury Planetary Orbiter (MPO) of the European Space Agency (ESA) space mission to Mercury named BepiColombo. The role of the accelerometer is to remove from the list of unknowns the non-gravitational accelerations that perturb the gravitational trajectory followed by the MPO in the strong radiation environment that characterises the orbit of Mercury around the Sun. Such a role is of fundamental importance in the context of the very ambitious goals of the Radio Science Experiments (RSE) of the BepiColombo mission. We have subdivided the errors on the accelerometer measurements into two main families: (i) the pseudo-sinusoidal errors and (ii) the random errors. The former are characterised by a periodic behaviour with the frequency of the satellite mean anomaly and its higher order harmonic components, i.e., they are deterministic errors. The latter are characterised by an unknown frequency distribution and we assumed for them a noise-like spectrum, i.e., they are stochastic errors. Among the pseudo-sinusoidal errors, the main contribution is due to the effects of the gravity gradients and the inertial forces, while among the random-like errors the main disturbing effect is due to the MPO centre-of-mass displacements produced by the onboard High Gain Antenna (HGA) movements and by the fuel consumption and sloshing. Very subtle to be considered are also the random errors produced by the MPO attitude corrections necessary to guarantee the nadir pointing of the spacecraft. We have therefore formulated the ISA error budget and the requirements for the satellite in order to guarantee an orbit reconstruction for the MPO spacecraft with an along-track accuracy of about 1 m over the orbital period of the satellite around Mercury in such a way to satisfy the RSE requirements.     

Impact of Limitations in Geophysical Background Models on Follow-on Gravity Missions

Purpose of this article is to demonstrate the effect of background geophysical corrections on a follow-on gravity mission. We investigate the quality of two effects, tides and atmospheric pressure variations, which both act as a surface load on the lithosphere. In both cases direct gravitational attraction of the mass variations and the secondary potential caused by the deformation of the lithosphere are sensed by a gravity mission. In order to assess the current situation we have simulated GRACE range-rate errors which are caused by differences in present day tide and atmospheric pressure correction models. Both geophysical correction models are capable of generating range-rate errors up to 10 μm/s and affect the quality of the recovered temporal and static gravity fields. Unlike missions such as TOPEX/Poseidon where tides can be estimated with the altimeter, current gravity missions are only to some degree capable of resolving these (geo)physical limitations. One of the reasons is the use of high inclination low earth orbits without a repeating ground track strategy. The consequence is that we will face a contamination of the gravity solution, both in the static and the time variable part. In the conclusions of this paper we provide suggestions for improving this situation, in particular in view of follow-on gravity missions after GRACE and GOCE, which claim an improved capability of estimating temporal variations in the Earth’s gravity field.     

GOCE卫星重力计划及其应用

基于CHAMP和GRACE卫星，GOCE(Gravity Field and Stead—state Ocean Circulation Explore)是欧空局(ESA)的一颗重力场和静态洋流探测卫星。利用它可得到空间分辨率为200—80km的全球重力场模型和1cm精度的大地水准面．简要介绍了目前重力卫星的发展现状与其局限性，详细叙述了GOCE卫星的组成、科学目标、测量原理、在地球物理等学科中的重要应用，并提出GOCE等重力卫星资料在我国的应用设想。     

Geoid and Gravity in Earth Sciences – An Overview

Precise global geoid and gravity anomaly information serves essentially three different kinds of applications in Earth sciences: gravity and geoid anomalies reflect density anomalies in oceanic and continental lithosphere and the mantle; dynamic ocean topography as derived from the combination of satellite altimetry and a global geoid model can be directly transformed into a global map of ocean surface circulation; any redistribution or exchange of mass in Earth system results in temporal gravity and geoid changes. After completion of the dedicated gravity satellite missions GRACE and GOCE a high standard of global gravity determination, both of the static and of the time varying field will be attained. Thus, it is the right time to investigate the future needs for improvements in the various fields of Earth sciences and to define the right strategy for future gravity field satellite missions.     

Determination of Shape, Gravity, and Rotational State of Asteroid 433 Eros

Prior to the Near Earth Asteroid Rendezvous (NEAR) mission, little was known about Eros except for its orbit, spin rate, and pole orientation, which could be determined from ground-based telescope observations. Radar bounce data provided a rough estimate of the shape of Eros. On December 23, 1998, after an engine misfire, the NEAR-Shoemaker spacecraft flew by Eros on a high-velocity trajectory that provided a brief glimpse of Eros and allowed for an estimate of the asteroid's pole, prime meridian, and mass. This new information, when combined with the ground-based observations, provided good a priori estimates for processing data in the orbit phase.After a one-year delay, NEAR orbit operations began when the spacecraft was successfully inserted into a 320×360 km orbit about Eros on February 14, 2000. Since that time, the NEAR spacecraft was in many different types of orbits where radiometric tracking data, optical images, and NEAR laser rangefinder (NLR) data allowed a determination of the shape, gravity, and rotational state of Eros. The NLR data, collected predominantly from the 50-km orbit, together with landmark tracking from the optical data, have been processed to determine a 24th degree and order shape model. Radiometric tracking data and optical landmark data were used in a separate orbit determination process. As part of this latter process, the spherical harmonic gravity field of Eros was primarily determined from the 10 days in the 35-km orbit. Estimates for the gravity field of Eros were made as high as degree and order 15, but the coefficients are determined relative to their uncertainty only up to degree and order 10. The differences between the measured gravity field and one determined from a constant density shape model are detected relative to their uncertainty only to degree and order 6. The offset between the center of figure and the center of mass is only about 30 m, indicating that Eros has a very uniform density (1% variation) on a large scale (35 km). Variations to degree and order 6 (about 6 km) may be partly explained by the existence of a 100-m, regolith or by small internal density variations. The best estimates for the J2000 right ascension and declination of the pole of Eros are α=11.3692±0.003° and δ=17.2273±0.006°. The rotation rate of Eros is 1639.38922±0.00015°/day, which gives a rotation period of 5.27025547 h. No wobble greater than 0.02° has been detected. Solar gravity gradient torques would introduce a wobble of at most 0.001°.     

Calibrating Data from the Hinode/X-Ray Telescope and Associated Uncertainties

The X-Ray Telescope (XRT) onboard the Hinode satellite, launched 23 September 2006 by the Japan Aerospace Exploration Agency (JAXA), is a joint mission of Japan, the United States, and the United Kingdom to study the solar corona. In particular, XRT was designed to study solar plasmas with temperatures between 1 and 10 MK with ≈?1″ pixels (≈?2″ resolution). Prior to analysis, the data product from this instrument must be properly calibrated and data values quantified to accurately assess the information contained within. We present here the standard methods of calibration for these data. The calibration was performed on an empirical basis that uses the least complicated correction that accurately describes the data while suppressing spurious features. By analyzing the uncertainties remaining in the data after calibration, we conclude that the procedure is successful, because the remaining uncertainty after calibration is dominated by photon noise. This calibration software is available in the SolarSoft software library.     

GRACE observations of changes in continental water storage

Signatures between monthly global Earth gravity field solutions obtained from GRACE satellite mission data are analyzed with respect to continental water storage variability. GRACE gravity field models are derived in terms of Stokes' coefficients of a spherical harmonic expansion of the gravitational potential from the analysis of gravitational orbit perturbations of the two GRACE satellites using GPS high–low and K-band low–low intersatellite tracking and on-board accelerometry. Comparing the GRACE observations, i.e., the mass variability extracted from temporal gravity variations, with the water mass redistribution predicted by hydrological models, it is found that, when filtering with an averaging radius of 750 km, the hydrological signals generated by the world's major river basins are clearly recovered by GRACE. The analyses are based on differences in gravity and continental water mass distribution over 3- and 6-month intervals during the period April 2002 to May 2003. A background model uncertainty of some 35 mm in equivalent water column height from one month to another is estimated to be inherent in the present GRACE solutions at the selected filter length. The differences over 3 and 6 months between the GRACE monthly solutions reveal a signal of some 75 mm scattering with peak values of 400 mm in equivalent water column height changes over the continents, which is far above the uncertainty level and about 50% larger than predicted by global hydrological models. The inversion method, combining GRACE results with the signal and stochastic properties of a hydrological model as ‘a priori’ in a statistical least squares adjustment, significantly reduces the overall power in the obtained water mass estimates due to error reduction, but also reflects the current limitations in the hydrological models to represent total continental water storage change in particular for the major river basins.     

Analytical and numerical study of the ground-track resonances of Dawn orbiting Vesta

The aim of Dawn mission is the acquisition of data from orbits around two bodies (4) Vesta and (1) Ceres, the two most massive asteroids.Due to the low thrust propulsion, Dawn will slowly cross and transit through ground-track resonances, where the perturbations on Dawn orbit may be significant. In this context, to safety go the Dawn mission from the approach orbit to the lowest science orbit, it is essential to know the properties of the crossed resonances.This paper analytically investigates the properties of the major ground-track resonances (1:1, 1:2, 2:3 and 3:2) appearing for Vesta orbiters: location of the equilibria, aperture of the resonances and period at the stable equilibria. We develop a general method using an averaged Hamiltonian formulation with a spherical harmonic approximation of the gravity field. If the values of the gravity field coefficient change, our method stays correct and applicable. We also discuss the effect of one uncertainty on the C₂₀ and C₂₂ coefficients on the properties of the 1:1 resonance. These results are checked by numerical tests. We determine that the increase of the eccentricity appearing in the 2:3 resonance is due to the C₂₂ and S₂₂ coefficients.Our method can be easily adapted to missions similar to Dawn because, contrarily to the numerical results, the analytical formalism stays the same and is valid for a wide range of physical parameters of the asteroid (namely the shape and the mass) as well as for different spacecraft orbits.Finally we numerically study the probability of the capture in resonance 1:1. Our paper reproduces, explains and supplements the results of Tricarico and Sykes (2010).     

Attitude stability of a spacecraft on a stationary orbit around an asteroid subjected to gravity gradient torque

Attitude stability of spacecraft subjected to the gravity gradient torque in a central gravity field has been one of the most fundamental problems in space engineering since the beginning of the space age. Over the last two decades, the interest in asteroid missions for scientific exploration and near-Earth object hazard mitigation is increasing. In this paper, the problem of attitude stability is generalized to a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid. This generalized problem is studied via the linearized equations of motion, in which the harmonic coefficients $C_{20}$ and $C_{22}$ of the gravity field of the asteroid are considered. The necessary conditions of stability of this conservative system are investigated in detail with respect to three important parameters of the asteroid, which include the harmonic coefficients $C_{20}$ and $C_{22}$ , as well as the ratio of the mean radius to the radius of the stationary orbit. We find that, due to the significantly non-spherical shape and the rapid rotation of the asteroid, the attitude stability domain is modified significantly in comparison with the classical stability domain predicted by the Beletskii–DeBra–Delp method on a circular orbit in a central gravity field. Especially, when the spacecraft is located on the intermediate-moment principal axis of the asteroid, the stability domain can be totally different from the classical stability domain. Our results are useful for the design of attitude control system in the future asteroid missions.     

Odyssey: a solar system mission

The Solar System Odyssey mission uses modern-day high-precision experimental techniques to test the laws of fundamental physics which determine dynamics in the solar system. It could lead to major discoveries by using demonstrated technologies and could be flown within the Cosmic Vision time frame. The mission proposes to perform a set of precision gravitation experiments from the vicinity of Earth to the outer Solar System. Its scientific objectives can be summarized as follows: (1) test of the gravity force law in the Solar System up to and beyond the orbit of Saturn; (2) precise investigation of navigation anomalies at the fly-bys; (3) measurement of Eddington’s parameter at occultations; (4) mapping of gravity field in the outer solar system and study of the Kuiper belt. To this aim, the Odyssey mission is built up on a main spacecraft, designed to fly up to 13 AU, with the following components: (a) a high-precision accelerometer, with bias-rejection system, measuring the deviation of the trajectory from the geodesics, that is also giving gravitational forces; (b) Ka-band transponders, as for Cassini, for a precise range and Doppler measurement up to 13 AU, with additional VLBI equipment; (c) optional laser equipment, which would allow one to improve the range and Doppler measurement, resulting in particular in an improved measurement (with respect to Cassini) of the Eddington’s parameter. In this baseline concept, the main spacecraft is designed to operate beyond the Saturn orbit, up to 13 AU. It experiences multiple planetary fly-bys at Earth, Mars or Venus, and Jupiter. The cruise and fly-by phases allow the mission to achieve its baseline scientific objectives [(1) to (3) in the above list]. In addition to this baseline concept, the Odyssey mission proposes the release of the Enigma radio-beacon at Saturn, allowing one to extend the deep space gravity test up to at least 50 AU, while achieving the scientific objective of a mapping of gravity field in the outer Solar System [(4) in the above list].      

Influence of geomagnetic field on the motion of low satellites

The first aim of the present work is to compute a more accurate and recent model for the Earth’s magnetic field. The second aim is to determine the effects of the Earth’s magnetic field on the motion of a charged artificial satellite to evaluate the variations of the orbital elements of the satellite due to these effects. The magnetic field and its variation with time have been studied at different heights, longitudes and latitudes. The geomagnetic field is considered as a multiple potential field and the electrical charge of the satellite is assumed to be constant. A new computer code has been constructed to follow the components of the magnetic field in spherical harmonic models. The Gauss equations are solved numerically. The results concentrate on the computation of the numerical values of orbital perturbation for the case of a low Earth satellite. RS-1 satellite and space craft gravity probe B (GPB) are chosen as cases of studies for a detailed numerical analysis.