Non-conservative forces on LAGEOS I and II

The behavior of the empirically determined along-track accelerations from the LAGEOS satellites has been attributed to a variety of physical phenomena. Of these, the models for Yarkovsky thermal drag, anisotropic reflectivity and Yarkovsky-Schach drag are dependent on the spin axis orientation of the satellite. This investigation explores the utilization of these models in an attempt to recover the spin axis orientation history, particularly for the LAGEOS I satellite.     

The LAGEOS satellites orbit and Yukawa-like interactions

LAGEOS II general relativity pericenter precession has been analysed in terms of the errors produced by the mismodelling of both the gravitational and non-gravitational perturbations acting on the satellite orbit. The accuracy in the pericenter determination may be considered as an upper-bound value for the estimate of the strength α of a possible new-long-range-interaction described by a Yukawa-like potential. In the present work we have focused on the constraints in α that can be obtained with the current best multi-satellites gravity field model EGM96 (α < 2.6 × 10⁻¹⁰) and also with the first promising models from the CHAMP (α < 1.8 × 10⁻¹⁰) and GRACE (α < 1.2 × 10⁻¹⁰) gravimetric missions. These results represent, potentially, an improvement of two or three orders-of-magnitude with respect to the best constraints obtained in the past with Earth–LAGEOS and Lunar–LAGEOS data (|α| < 10⁻⁵–10⁻⁸). The impact of the non-gravitational perturbations mismodelling in the final error budget has been determined together with the improvements obtainable in the constraint of the strength α with the proposed LARES satellite.     

Determination of station positions and velocities from laser ranging observations to Ajisai,Starlette and Stella satellites

The positions and velocities of the four Satellite Laser Ranging (SLR) stations: Yarragadee (7090), Greenbelt (7105), Graz (7839) and Herstmonceux (7840) from 5-year (2001–2005) SLR data of low orbiting satellites (LEO): Ajisai, Starlette and Stella were determined. The orbits of these satellites were computed from the data provided by 20 SLR stations. All orbital computations were performed by means of NASA Goddard’s GEODYN-II program. The geocentric coordinates were transformed to the topocentric North–South, East–West and Vertical components in reference to ITRF2005. The influence of the number of normal points per orbital arc and the empirical acceleration coefficients on the quality of station coordinates was studied. To get standard deviation of the coordinates determination lower than 1 cm, the number of the normal points per site had to be greater than 50. The computed positions and velocities were compared to those derived from LAGEOS-1/LAGEOS-2 data. Three parameters were used for this comparison: station coordinates stability, differences from ITRF2005 positions and velocities. The stability of coordinates of LEO satellites is significantly worse (17.8 mm) than those of LAGEOS (7.6 mm), the better results are for Ajisai (15.4 mm) than for Starlette/Stella (20.4 mm). The difference in positions between the computed values and ITRF2005 were little bit worse for Starlette/Stella (6.6 mm) than for LAGEOS (4.6 mm), the results for Ajisai were five times worse (29.7 mm) probably due to center of mass correction of this satellite. The station velocities with some exceptions were on the same level (≈1 mm/year) for all satellites. The results presented in this work show that results from Starlette/Stella are better than those from Ajisai for station coordinates determination. We can applied the data from LEO satellites, especially Starlette and Stella for determination of the SLR station coordinates but with two times lower accuracy than when using LAGEOS data.     

Thermal-optical design of a geodetic satellite for one millimeter accuracy

The LAGEOS (LAser GEOdynamic Satellite) satellites use a 1.5 in. uncoated retroreflector (cube corner). Design studies done for LAGEOS-1 showed that using smaller cubes would result in greater accuracy and lower thermal gradients. However, this would require using a larger number of cubes. Simulations showed that the accuracy goal of 5 mm could be met using 1.5 in. cubes by adding a dihedral angle offset. The LARES (LAser RElativity Satellite) satellite launched in 2012 is a smaller version of LAGEOS using the same size cube corner and floating mount as LAGEOS.The recent development of COTS (Commercial Off-The-Shelf) cube corners has eliminated cost as an obstacle to using a larger number of smaller cubes. COTS cubes have no dihedral angle offset. However, no offset is needed if the size is chosen properly. The diffraction pattern of a 1.0 in. uncoated cube with no dihedral angle offset has 6 lobes around the central peak due to total internal reflection, The velocity aberration for LAGEOS is about 32–40 microradians. The OCS (Optical Cross Section) of a one inch uncoated COTS retroreflector is about 0.5 million sq m for the LAGEOS orbit.Testing of 10 inexpensive COTS cubes by Ludwig Grunwaldt and Reinhart Neubert shows good cross section (unpublished work done at GFZ Potsdam, Germany). Measurement of 50 COTS cubes at INFN (Mondaini et al., 2018), shows a loss of cross section of only 33% (Slide 10). Simulations show that systematic range errors on the order of a half millimeter are possible for a spherical geodetic satellite such as LARES. Adjustments for the holding and ejection system result in some loss of accuracy.     

Characteristics and accuracies of the GRACE inter-satellite pointing

For almost 10 years, the Gravity Recovery and Climate Experiment (GRACE) has provided information about the Earth gravity field with unprecedented accuracy. Efforts are ongoing to approach the GRACE baseline accuracy as there still remains an order of magnitude between the present error level of the gravity field solutions and the GRACE baseline. At the current level of accuracy, thorough investigation of sensor related effects is necessary as they are one of the potential contributors to the error budget. In the science mode operations, the twin satellites are kept precisely pointed with their KBR antennas towards each other. It is the task of the onboard attitude and orbit control system (AOCS) to keep the satellites in the required formation. We analyzed long time series of the inter-satellite pointing variations as they reflect the AOCS performance and characteristics. We present significant systematic effects in the inter-satellite pointing and discuss their possible sources. Prominent features are especially related to the magnetic torquer characteristics, star cameras’ performance and KBR antenna calibration parameters. The relation between the magnetic torquer attitude control and the Earth magnetic field, impact of the different performance of the two star camera heads on the attitude control and the features due to uncertainties in the calibration parameters relating the star camera frame to K-frame are discussed in detail. Proper understanding of these effects will help to reduce their impact on the science data and subsequently increase the accuracy of the gravity field solutions. Moreover, understanding the complexity of the onboard system is essential not only for increasing the accuracy of the GRACE data but also for the development of the future gravity field satellite missions.     

Estimation of the elastic Earth parameters (h2, l2) using SLR data

We present results for the global elastic parameters h₂ and l₂ derived from the analysis of Satellite Laser Ranging (SLR) data. SLR data for the two satellites LAGEOS 1 and LAGEOS 2 observed during 2.5 years from January 3, 2005 until July 1, 2007 with 18 globally distributed ground stations were analysed using different approaches. The analysis was done separately for the two satellites and approaches to estimate the two elastic parameters independently and together were performed. We do a sequential analysis and study the stability of the estimates as a function of length of the data set used. The adjusted final values for h₂ equal to 0.6151 ± 0.0008 and 0.6152 ± 0.0008, and those for l₂ equal to 0.0886 ± 0.0003 and 0.0881 ± 0.0003 for LAGEOS 1 and LAGEOS 2 tracking data are compared to other independently derived estimates. These parameters and their errors achieve stability at about the 24 and 27 month time interval for h₂ and l₂, respectively.     

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.     

Radiative forces and LAGEOS' orbit

The orbit analysis of LAGEOS satellite has resulted into the discovery and/or reassessment of several very small sources of perturbation on satellite orbits. The analysis of orbital arcs of duration ranging from one month to several years has revealed that perturbative effects are present, having unpredicted long-term or secular components down to the 10⁻¹² m/s² acceleration level. It was soon realized that those perturbations have a non-gravitational origin.

In recent years, we have devoted some effort to the physical modelling of radiative perturbations, caused by momentum exchanges with an appropriate radiative field, and have considered their potential role in the analysis of LAGEOS orbit residuals. These perturbations include: (i) direct solar radiation pressure; (ii) radiation pressure from the Earth's emitted/reflected/diffused radiation flux; (iii) the so-called thermal thrust force.

The main results of this work are reviewed, discussing its relationships with models developed by other research groups. In particular, we present a list of the physical processes which still appear to need more detailed and realistic modelling to reach a better understanding of LAGEOS dynamics at the 10⁻¹² m/s² level.     

Finding the suitable drag-free acceleration noise level for future low-low satellite-to-satellite tracking geodesy missions

This paper evaluates the impact of residual acceleration noise on the estimation of the Earth’s time-varying gravity field for future low-low satellite-to-satellite tracking missions. The goal is to determine the maximum level of residual acceleration noise that does not adversely affect the estimation error. The Gravity Recovery And Climate Experiment (GRACE) has provided monthly average gravity field solutions in spherical harmonic coefficients for more than a decade. It provides information about land and ocean mass variations with a spatial resolution of ～350?km and with an accuracy within 2?cm throughout the entire Earth. GRACE Follow-on was launched in May 2018 to advance the work of GRACE and to test a new laser ranging interferometer, which measures the range between the two satellites with higher precision than the K-Band ranging system used in GRACE. Moreover, there have been simulation studies that show, an additional pair of satellites in an inclined orbit increases the sampling frequency and reduces temporal aliasing errors. Given the fact that future missions will likely continue to use the low-low satellite-to-satellite tracking formation with laser ranging interferometry, it is expected that the residual acceleration noise will become one of the largest error contributor for the time-variable gravity field solution. We evaluate three different levels of residual acceleration noise based on demonstrated drag-free systems to find a suitable drag-free performance target for upcoming geodesy missions. We analyze both a single collinear polar pair and the optimal double collinear pair of drag-free satellites and assume the use of a laser ranging interferometer. A partitioned best linear unbiased estimator that was developed, incorporating several novel features from the ground up is used to compute the solutions in terms of spherical harmonics. It was found that the suitable residual acceleration noise level is around 2?×?10^?12?ms^?2?Hz^?1/2. Decreasing the acceleration noise below this level did not result in more accurate gravity field solutions for the chosen mission architecture.     

SPOT-2 and TOPEX/Poseidon precise orbit determination from DORIS doppler tracking

The French earth observation satellite SPOT-2 has served as a testbed for precise orbit determination from DORIS doppler tracking in anticipation of the TOPEX/Poseidon mission. Using the most up-to-data gravity field model, JGM-2, a radial orbit accuracy of about 2–9 cm was achieved, with an rms of fit of the tracking data of about 0.64 mm/s. Furthermore, it was found that the coordinates of the ground stations can be determined with an accuracy of the order of 2–5 cm after removal of common rotations, and translations.

Using a slightly different model for atmospheric drag, but the same gravity model, precise orbits of TOPEX/Poseidon from DORIS tracking data were determined with a radial orbit accuracy of the order of 4–5 cm, which is far within the 13 cm mission requirement. This conclusion is based on the analysis of 1-day overlap of successive 11-day orbits, and the comparisons with orbits computed from satellite laser tracking (SLR) and from the combination of SLR and DORIS tracking. Results indicate a consistency between the different orbits of 1–4 cm, 4–20 cm, and 6–13 cm in the radial, cross-track, and along-track directions, respectively. The residual rms is about 4–5 cm for SLR data and 0.56 mm/s for DORIS tracking. These numbers are roughly twice as large as the system noise levels, reflecting the fact that there are still some modeling errors left.     

基于会切场推力器无拖曳系统建模及参数优化

随着卫星重力测量技术的突破性进展,对航天器试验环境要求也在不断提高,航天器受到的残余扰动必须尽可能减小。作为中国将来重力场测量卫星备选主推力器的会切场推力器,其推力器的控制精度直接决定了测量的准确性。文章首先通过PID方法设计了位移模式下的无拖曳控制器,该控制器在预估阻力系数、参考质量与卫星本体的位移差、速度差等性能方面有良好的表现,在应对卫星运行时的突发情况时表现出很强的稳定性。但PID参数没有达到最优解,在此基础上对于该模型的控制精度进行优化,用遗传算法对PID控制的参数进行筛选。结果分析表明,会切场推力器的控制精度有所改善,NTW方向上的速度和位移误差均减小;推力阻力和显著减少;控制精度提高,更好地满足使用需求。     

Influence of time variable geopotential models on precise orbits of altimetry satellites,global and regional mean sea level trends

During the last decade a significant progress has been reached in the investigation of the gravity field of the Earth. Besides static, also time variable geopotential models have been recently created. In this paper we investigate the impact of the recent time variable geopotential models on altimetry satellite orbits and such altimetry products based on these orbits, as global and regional mean sea level trends. We show that the modeling of time variable gravity improves the orbit solutions, at least for the GRACE period where time variable gravity is sufficiently accurately observed by this mission. Our analysis includes six geopotential models jointly developed by GFZ German Research Centre for Geosciences and Space Geodesy Research Group (CNES/GRGS) Toulouse: the stationary model EIGEN-GL04S, a stationary version of EIGEN-6S (EIGEN-6S_stat), a corrected version of EIGEN-6S and three enhanced versions of EIGEN-6S called EIGEN-6S2, EIGEN-6S2A and EIGEN-6S2B. By “stationary” we mean “containing periodic parameters such as annual and semi-annual variations, but no secular (drift) terms”. We computed precise orbits for the radar altimetry satellites ERS-1, ERS-2, TOPEX/Poseidon, and Envisat over 20 years between 1991 and 2011. The orbit, single-mission and multi-mission altimetry crossover analyses show that the time variable models EIGEN-6S_corrected, EIGEN-6S2 and its two precursors EIGEN-6S2A/B perform notably better than the stationary models for the GRACE period from 2003 onwards. Thus, using EIGEN-6S2 and EIGEN-6S2A/B we have got 3.6% smaller root mean square fits of satellite laser ranging observations for Envisat, as when using EIGEN-GL04S. However, for the pre-GRACE period 1991–2003, the stationary geopotential models EIGEN-GL04S and EIGEN-6S_stat as well as EIGEN-6S2 having no drift terms for degree 3–50 at this time interval perform superior compared to EIGEN-6S_correct and EIGEN-6S2A/B which contain drifts for this period. We found, that the time variable geopotential models have a low (0.1–0.2 mm/yr) impact on our results for the global mean sea level trend. However, we found strong East/West differences up to 3 mm/yr in the regional mean sea level trends when using orbits of all four satellites based on time variable and stationary geopotential models. We show that these differences are related to the relative drifts of the centers-of-origin between the orbit solutions based on the time variable and stationary geopotential models. From the results of our detailed study, we conclude that the final version of the time variable gravity field model EIGEN-6S2 performs best for the four satellites tested. This model provides the most reliable and mission-consistent sea level estimates for the whole time period from 1992 to 2010. This model is of maximum spherical harmonic degree and order 260 and contains time series for drifts as well as annual and semiannual variations of the spherical harmonic coefficients for degree 2–50.     

Precise orbit determination and sea-surface topography from ERS-1

Results of ERS - 1 orbit refinement procedures for long-arcs are summarised with particular reference to gravity field modelling and sea-surface topography. Long arcs from the three day and thirty - five day repeat cycles have been used to refine the earth's gravity field constrained by the JGM- 2 covariance matrix and simultaneously to solve for the quasi - stationary sea - surface topography. Orbital computations are presented for arcs not included within the refinement process with improvements identified through rms of fit to laser, altimeter and single satellite crossover data. Additional results utilising dual crossovers with TOPEX / POSEIDON in a full dynamic process are also presented. In particular, reference is made to the relative altimeter biases between the respective satellites and the characteristics of the orbital adjustment through inclusion of dual crossover 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.     

Estimating low resolution gravity fields at short time intervals to reduce temporal aliasing errors

The Gravity Recovery and Climate Experiment (GRACE) satellite mission has been estimating temporal changes in the Earth’s gravitational field since its launch in 2002. While it is not yet fully resolved what the limiting source of error is for GRACE, studies on future missions have shown that temporal aliasing errors due to undersampling signals of interest (such as hydrological variations) and errors in atmospheric, ocean, and tide models will be a limiting source of error for missions taking advantage of improved technologies (flying drag-free with a laser interferometer). This paper explores the option of reducing the effects of temporal aliasing errors by directly estimating low degree and order gravity fields at short time intervals, ultimately resulting in data products with improved spatial resolution. Three potential architectures are considered: a single pair of polar orbiting satellites, two pairs of polar orbiting satellites, and a polar orbiting pair of satellites coupled with a lower inclined pair of satellites. Results show that improvements in spatial resolution are obtained when one estimates a low resolution gravity field every two days for the case of a single pair of satellites, and every day for the case of two polar pairs of satellites. However, the spatial resolution for these cases is still lower than that provided by simply destriping and smoothing the solutions via standard GRACE post-processing techniques. Alternately, estimating daily gravity fields for the case of a polar pair of satellites coupled with a lower inclined pair results in solutions with superior spatial resolution than that offered by simply destriping and smoothing the solutions.     

Mapping the earth's magnetic and gravity fields from space: Current status and future prospects

Orbital potential field measurements are sensitive to regional variations in earth density and magnetization that occur over scales of a few hundred kilometers or greater. Global field models currently available are able to distinguish gravity variations of ±5 milligal over distances of ～1,000 km and magnetic variations of ±6 gamma over distances of ～300 km at the earth's surface. Regional variations in field strength have been detected in orbital measurements that are not apparent in higher resolution, low altitude surveys. NASA is presently studying a spacecraft mission known as GRAVSAT/MAGSAT, which would be the first satellite mission to perform a simultaneous survey of the earth's gravity and magnetic fields at low orbital altitudes. GRAVSAT/MAGSAT has been proposed for launch during the latter nineteen-eighties, and it would measure gravity field strength to an accuracy of 1 milligal and magnetic field strength to an accuracy of 2 gamma (scalar)/5 gamma (vector components) over a distance of roughly 100 km. Even greater improvements in the accuracy and spatial resolution of orbital surveys are anticipated during the nineteen-nineties with the development of potential field gradiometers and a tethered satellite system that can be deployed from the Space Shuttle to altitudes of 120 km above the earth's surface.     

Dynamic and non-dynamic ERS-1 radial orbit improvement from ERS-1/TOPEX dual-satellite altimetry

Dual-satellite altimeter crossover differences between ERS-1 and TOPEX/Poseidon have been included as supplementary tracking data in ERS-1 orbit computations from SLR and single-satellite crossover differences. It was found that including the dual-satellite crossover differences slightly improves the ERS-1 radial orbit accuracy of about 12 cm for orbits computed with the JGM-2 gravity field and also leads to a better ‘centering’ of the ERS-1 orbit in the terrestrial reference frame defined for TOPEX/Poseidon. In addition to this dynamic orbit improvement technique, a non-dynamic technique has been investigated that removes the larger part of the ERS-1 radial orbit error from the dual-satellite crossover difference residuals. For ERS-1 orbits computed with the GEM-T2 gravity field, it was found that the non-dynamic technique could improve the radial orbit accuracy from 140 cm to the same level of accuracy as the ERS-1 JGM-2 orbits.     

The temporal variation of mean gravity of atmosphere

One of the precise widely used global Zenith Hydrostatic Delay (ZHD) model is based on the gravity value at the centroid of the atmospheric column at the station of observation and gravity value at the centroid is constant in this model for a specific location throughout the year. However, as the content and extent of atmosphere varies temporally, its centroid and consequently gravity value at the centroid also varies. Apart from this, the actual atmospheric condition of different region is not alike. Therefore, there is a need to develop a regional mean gravity model and development of such model has been discussed in this paper. To obtain the mean gravity model, first a regional model of centroid height of atmospheric column was developed as a function of the surface pressure and temperature. It was developed by multiple regressions between estimated centroid of the atmosphere and surface pressure, surface temperature using radiosonde data of five radiosonde stations spread over the Indian subcontinent. Three years radiosonde data from 2006 to 2008 was used for each station. The root mean square error in estimating centroid of the atmospheric column is about ±326 m, which is negligible considering the variability of the atmosphere and its content. The centroid height model has been used to formulate the mean gravity model, considering uniform lapse rate in gravity with height. It is found that proposed mean gravity model provides temporal variation of mean gravity values at the centroid and thus matches with the reality. The interesting advantage of the developed model is that the model shows diurnal variation of mean gravity. The accuracy of ZHD has shown of the order of about 0.3 mm using the developed regional mean gravity model. However, already developed ZHD model has shown a slight inferior result compared to the developed model. These models have shown accuracy of about 0.8 mm and 0.6 mm.     

Generation of vertical–horizontal and horizontal–horizontal gravity gradients using stochastically modified integral estimators

The Earth’s gravity field modelling is an ill-posed problem having a sensitive solution to the error of data. Satellite gravity gradiometry (SGG) is a space technique to measure the second-order derivatives of geopotential for modelling this field, but the measurements should be validated prior to use. The existing terrestrial gravity anomalies and Earth gravity models can be used for this purpose. In this paper, the second-order vertical–horizontal (VH) and horizontal–horizontal (HH) derivatives of the extended Stokes formula in the local north-oriented frame are modified using biased, unbiased and optimum types of least-squares modification. These modified integral estimators are used to generate the VH and HH gradients at 250 km level for validation purpose of the SGG data. It is shown that, unlike the integral estimator for generating the second-order radial derivative of geopotential, the system of equations from which the modification parameters are obtained is unstable for all types of modification, with large cap size and high degree, and regularization is strongly required for solving the system. Numerical studies in Fennoscandia show that the SGG data can be estimated with an accuracy of 1 mE using an integral estimator modified by a biased type least-squares modification. In this case an integration cap size of 2.5° and a degree of modification of 100 for integrating 30′ × 30′ gravity anomalies are required.     

关于北斗卫星导航系统的被动式定位算法比较研究

我国北斗卫星导航系统空间卫星共有2或3颗,无法单独满足被动导航定位的要求．针对这种卫星稀少的情况提出了3种被动式定位算法: 2星定位算法、 3星3参数定位算法和3星4参数定位算法, 这些算法分别采用气压测高方法增加观测数据和采用数学模型描述接收机钟差的方法减少定位方程求解的未知数;探讨了北斗卫星导航系统备份星的可用性和对导航定位精度的贡献;还提出了准差分修正技术,提高了定位精度．实验证明, 3种算法都取得了100m以内的定位结果,可以满足一般用户定位需求．