A new method to determine the gravity field of small bodies from line-of-sight acceleration data

We present a new method to derive line-of-sight acceleration observables from spacecraft radio tracking data. The observables can be used to estimate the mass and gravity of a natural satellite as a spacecraft flyby. The corresponding observation model adapts to one-way and two/three-way tracking modes. As a test case for method validation and application, we estimated the mass and degree two gravity field for the Martian moon Phobos using simulated tracking data when the spacecraft Mars Express flew by Phobos on 2013 December 29. We have a few real tracking data during flyby and they will be used to confirm raw data simulation. The main purpose of this paper is to demonstrate the method of line-of-sight acceleration reduction from raw tracking data and the feasibility to estimate mass and gravity of a natural satellite using this type of observable. This novel method is potentially applicable to planet and asteroid gravity field studies combined with Doppler tracking data.     

Numerical simulations of a Mars geodesy network experiment: Effect of orbiter angular momentum desaturation on Mars' rotation estimation

The scientific objectives of a geodetic experiment based on a network of landers, such as NEIGE (NEtlander Ionosphere and Geodesy Experiment) are to improve the current knowledge of Mars' interior and atmosphere dynamics. Such a network science experiment allows monitoring the motions of the Martian rotation axis with a precision of a few centimeters (or milli-arc-seconds (mas)) over annual and sub-annual periods. Thereto, besides radio tracking of a Mars orbiter from the Earth, radio Doppler shifts between this orbiter and several landers at the planet's surface will be performed. From the analysis of these radio Doppler data, it is possible to reconstruct the orbiter motion and Mars' orientation in space. The errors on the orbit determination (position and velocity of the orbiter) have an impact on the geodetic parameters determination from the Doppler shifts and must be removed from the signal in order to achieve a high enough accuracy. In this paper, we perform numerical simulations of the two Doppler signals involved in such an experiment to estimate the impact of the spacecraft angular momentum desaturations on the determination of Mars' orientation variations. The attitude control of the orbiter needs such desaturation maneuvers regularly repeated. They produce velocity variations that must be taken into account when determining the orbit. For our simulations, we use a priori models of the Martian rotation and introduce the spacecraft velocity variations induced by each desaturation event. By a least-squares adjustment of the simulated Doppler signals, we then estimate the orbiter velocity variations and the parameters of the Mars' rotation model. We show that these velocity variations are ill resolved when the spacecraft is not tracked, therefore requiring a near-continuous tracking from the Earth to accurately determine the orbit. In such conditions we show that only 15- of lander-orbiter tracking per week allows recovering Mars' orientation parameters with a precision of a few mas over a period of 1 Martian year.     

Lander radioscience for obtaining the rotation and orientation of Mars

The paper presents the concept, the objectives, the approach used, and the expected performances and accuracies of a radioscience experiment based on a radio link between the Earth and the surface of Mars. This experiment involves radioscience equipment installed on a lander at the surface of Mars. The experiment with the generic name lander radioscience (LaRa) consists of an X-band transponder that has been designed to obtain, over at least one Martian year, two-way Doppler measurements from the radio link between the ExoMars lander and the Earth (ExoMars is an ESA mission to Mars due to launch in 2013). These Doppler measurements will be used to obtain Mars’ orientation in space and rotation (precession and nutations, and length-of-day variations). More specifically, the relative position of the lander on the surface of Mars with respect to the Earth ground stations allows reconstructing Mars’ time varying orientation and rotation in space.Precession will be determined with an accuracy better by a factor of 4 (better than the 0.1% level) with respect to the present-day accuracy after only a few months at the Martian surface. This precession determination will, in turn, improve the determination of the moment of inertia of the whole planet (mantle plus core) and the radius of the core: for a specific interior composition or even for a range of possible compositions, the core radius is expected to be determined with a precision decreasing to a few tens of kilometers.A fairly precise measurement of variations in the orientation of Mars’ spin axis will enable, in addition to the determination of the moment of inertia of the core, an even better determination of the size of the core via the core resonance in the nutation amplitudes. When the core is liquid, the free core nutation (FCN) resonance induces a change in the nutation amplitudes, with respect to their values for a solid planet, at the percent level in the large semi-annual prograde nutation amplitude and even more (a few percent, a few tens of percent or more, depending on the FCN period) for the retrograde ter-annual nutation amplitude. The resonance amplification depends on the size, moment of inertia, and flattening of the core. For a large core, the amplification can be very large, ensuring the detection of the FCN, and determination of the core moment of inertia.The measurement of variations in Mars’ rotation also determines variations of the angular momentum due to seasonal mass transfer between the atmosphere and ice caps. Observations even for a short period of 180 days at the surface of Mars will decrease the uncertainty by a factor of two with respect to the present knowledge of these quantities (at the 10% level).The ultimate objectives of the proposed experiment are to obtain information on Mars’ interior and on the sublimation/condensation of CO₂ in Mars’ atmosphere. Improved knowledge of the interior will help us to better understand the formation and evolution of Mars. Improved knowledge of the CO₂ sublimation/condensation cycle will enable better understanding of the circulation and dynamics of Mars’ atmosphere.     

由面波快速稳健反演地震矩张量的经验方法

给出一种稳健反演方法(也称迭代加权最小二乘IRLS),该方法对分离数据点极不敏感,因为它可以自动剔除异常的数据点,不需要人工仔细检查和控制数据质量。为了使用Tarantola-Valette的迭代广义离散反演方法进行稳健反演,将基于残差和信噪比的不同判据引入到协方差矩阵(作用相当于权重函数)中。在实际应用中,这个算法的作用相当于震源机制初步测定(PDFM),后者在海啸预警中用于快速估计强震震源参数。要反演的输入数据是长周期面波的频谱,输出的计算结果为地震矩张量,根据这些张量获得地震的震源几何形状和主应力轴。这种算法可以应用于任何非线性(迭代)反演计算中,解决污染数据集的分离数据点的问题。     

The precise positioning of lunar farside lander using a four-way lander-orbiter relay tracking mode

China is planning to land a spacecraft on the farside of the Moon, a premiere, by 2018. In essence, the traditional tracking modes, based on direct visibility, cannot operate for the lunar farside lander tracking, and therefore a relay satellite, visible at the same time by both the lander and the Earth, will be required, operating in the so-called four-way mode (Earth-relay satellite-lander-relay satellite-Earth). In this paper, we firstly give the mathematical formulation of the four-way relay tracking mode and of its partial derivatives with respect to the relevant parameters, implemented in our POD software WUDOGS (Wuhan University Deep-space Orbit determination and Gravity recovery System). In a second step, in simulation mode, we apply this relay mode to determining lander coordinates, which are absolutely needed for a sample return mission, or to add constraints on rotation models of the Moon. The results show that with Doppler measurements at a 0.1 mm/s error level, the positioning of the farside lander could be done at centimeters level (1-\(\delta\)) in the case of a circumlunar relay satellite; and at a 5 meters level (1-\(\delta\)) in the case of a Lagrange point (L2) Halo relay satellite.     

Ellipsoidal Harmonic expansions of the gravitational potential: Theory and application

Small bodies of the solar system are now the targets of space exploration. Many of these bodies have elongated, non-spherical shapes, and the usual spherical harmonic expansions of their gravity fields are not well suited for the modelling of spacecraft orbits around these bodies. An elegant remedy is to use ellipsoidal harmonic expansions instead of the usual spherical ones. In this paper, we present their mathematical theory as well as a real application: the simulation of a landing on the surface of a kilometer-sized comet. We show that with an ellipsoidal harmonic expansion up to degree 5, the error on the landing position is at the meter level, while the corresponding error for the spherical harmonic expansion can reach tens of meters.This revised version was published online in October 2005 with corrections to the Cover Date.     

A simulation experiment on the GM estimation for Comet 133P/Elst-Pizarro

In China’s asteroid mission to be launched around 2025,(7968)133 P/Elst-Pizarro(hereafter 133 P)will be the second target,after a visit to asteroid(469219)Kamo’oalewa.This paper describes a simulation of precise orbit determination for the spacecraft around comet 133 P,as well as estimation of its gravitational parameter(GM)value and the solar radiation pressure coefficient Cr for the spacecraft.Different cometocentric distances of 200,150 and 100 km orbits are considered,as well as two tracking modes:exclusive two-way range-rate mode(Earth station to spacecraft)and combinations of two-way range-rate and local spacecraft onboard ranging to the comet.Compared to exclusive two-way range-rate,the introduction of local ranging observables improves the final GM uncertainties by up to one order of magnitude.An ephemeris error in the orbit of 133 P is also considered,and we show that,to obtain a reliable estimate of the GM for 133 P,this error cannot exceed a one km range.     

Simulation of gravity gradients: a comparison study

At different European institutes software has been developed for evaluation of the gravitational potential of the Earth using high degree spherical harmonic expansions. In this report the results of a comparison of a number of these software packages are presented. We compared the results for the second order derivatives (gravity gradients). It appeared that one of the most critical points in these computations is the definition of the coordinates, which should be as accurate as possible. Machine dependency and algorithm setup were of less importance, the former being only reflected in CPU timing results.