Light-time computations for the BepiColombo Radio Science Experiment

The Radio Science Experiment is one of the on board experiments of the Mercury ESA mission BepiColombo that will be launched in 2014. The goals of the experiment are to determine the gravity field of Mercury and its rotation state, to determine the orbit of Mercury, to constrain the possible theories of gravitation (for example by determining the post-Newtonian parameters), to provide the spacecraft position for geodesy experiments and to contribute to planetary ephemerides improvement. This is possible thanks to a new technology which allows to reach great accuracies in the observables range and range rate; it is well known that a similar level of accuracy requires studying a suitable model taking into account numerous relativistic effects. In this paper we deal with the modelling of the space-time coordinate transformations needed for the light-time computations and the numerical methods adopted to avoid rounding-off errors in such computations.     

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.     

Global MHD modeling of Mercury's magnetosphere with applications to the MESSENGER mission and dynamo theory

We use a global magnetohydrodynamic (MHD) model to simulate Mercury's space environment for several solar wind and interplanetary magnetic field (IMF) conditions in anticipation of the magnetic field measurements by the MESSENGER spacecraft. The main goal of our study is to assess what characteristics of the internally generated field of Mercury can be inferred from the MESSENGER observations, and to what extent they will be able to constrain various models of Mercury's magnetic field generation. Based on the results of our simulations, we argue that it should be possible to infer not only the dipole component, but also the quadrupole and possibly even higher harmonics of the Mercury's planetary magnetic field. We furthermore expect that some of the crucial measurements for specifying the Hermean internal field will be acquired during the initial fly-bys of the planet, before MESSENGER goes into orbit around Mercury.     

Addressing some critical aspects of the BepiColombo MORE relativity experiment

The Mercury Orbiter Radio science Experiment (MORE) is one of the experiments on-board the ESA/JAXA BepiColombo mission to Mercury, to be launched in October 2018. Thanks to full on-board and on-ground instrumentation performing very precise tracking from the Earth, MORE will have the chance to determine with very high accuracy the Mercury-centric orbit of the spacecraft and the heliocentric orbit of Mercury. This will allow to undertake an accurate test of relativistic theories of gravitation (relativity experiment), which consists in improving the knowledge of some post-Newtonian and related parameters, whose value is predicted by General Relativity. This paper focuses on two critical aspects of the BepiColombo relativity experiment. First of all, we address the delicate issue of determining the orbits of Mercury and the Earth–Moon barycenter at the level of accuracy required by the purposes of the experiment and we discuss a strategy to cure the rank deficiencies that appear in the problem. Secondly, we introduce and discuss the role of the Solar Lense–Thirring effect in the Mercury orbit determination problem and in the relativistic parameters estimation.     

Simultaneous determination of global topography, tidal Love number and libration amplitude of Mercury by laser altimetry

Solar tidal forces generate elevation changes of Mercury's surface of the order 1 m within one Hermean year, and solar torques on the non-symmetric permanent mass distribution of the planet cause an uneven rotation of Mercury's surface with a libration amplitude of the order of 40 arcsec. Knowledge of the precise reaction of the planet to tidal forcing, expressed by the Love numbers h₂ and k₂, as well as accurate knowledge of the amplitude of forced libration Φ_lib, puts constraints on the internal structure, for example the state and the size of the core. The MESSENGER and BepiColombo missions to Mercury carry laser altimeters, whose primary goal is to accurately map the topography. Here we investigate if the Love number h₂ and the amplitude of forced libration can be determined together with the static topography of the planet from a global altimetry record. We do this by creating synthetic altimeter data for the nominal orbit of BepiColombo over the nominal mission duration of approximately four Mercury years and inverting them for the static and time-dependent parts of the topography. We assume purely Gaussian noise. We find that it is possible to extract both parameters h₂ and Φ_lib with an accuracy of approximately 10%, while the static topography coefficients of a spherical harmonic expansion can be determined simultaneously with an accuracy at the centimetre level. Extraction of the static topography to higher harmonic degrees improves the precision of the measurement of h₂ and Φ_lib. The simulation results demonstrate that it seems feasible to test current models on Mercury's interior with sufficient precision using BepiColombo Laser Altimeter data.     

Simulation of precise orbit determination of lunar orbiters

Based on the ongoing Chinese lunar exploration mission, i.e. the “Chang'e 1” project, precise orbit determination of lunar orbiters is analyzed for the actual geographical distribution and observational accuracy of the Chinese united S-band (USB) observation and control network as well as the very long baseline interferometry (VLBI) tracking network. The observed data are first simulated, then solutions are found after including the effects of various error sources and finally compared. We use the space data analysis software package, GEODYN, developed at Goddard Space Flight Center, NASA, USA. The primary error source of the flight orbiting the moon is the lunar gravity field. Therefore, the (formal) error of JGL165P1, i.e. the model of the lunar gravity field with the highest accuracy at present, is first discussed. After simulating the data of ranging and velocity measurement as well as the VLBI data of the time delay and time delay rate, precise orbit determination is carried out when the error of the lunar gravity field is added in. When the orbit is determined, the method of reduced dynamics is adopted with the selection of appropriate empirical acceleration parameters to absorb the effect of errors in the lunar gravity field on the orbit determination. The results show that for lunar missions like the “Chang'e 1” project, that do not take the lunar gravity field as their main scientific objective, the method of reduced dynamics is a simple and effective means of improving the accuracy of the orbit determination of the lunar orbiters.     

Mercury's rotation axis and period

Recent measurements made from high-resolution Mariner 10 photography of the planet Mercury yield a rotation period of 58.6461 ± 0.005 days, in excellent agreement with the period required for a precise $\text{2}\text{3}$ resonance with its orbital period (58.6462 days). The axis of rotation of the planet was calculated to be offset about 2° from the perpendicular to its orbital plane within a 50% probability error ellipse of ±2.6° by ±6.5°. Dynamical considerations make it most likely that the true displacement from the orbit normal is less than 1°.     

The MESSENGER mission to Mercury: scientific payload

The MErcury, Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) mission will send the first spacecraft to orbit the planet Mercury. A miniaturized set of seven instruments, along with the spacecraft telecommunications system, provide the means of achieving the scientific objectives that motivate the mission. The payload includes a combined wide- and narrow-angle imaging system; γ-ray, neutron, and X-ray spectrometers for remote geochemical sensing; a vector magnetometer; a laser altimeter; a combined ultraviolet-visible and visible-infrared spectrometer to detect atmospheric species and map mineralogical absorption features; and an energetic particle and plasma spectrometer to characterize ionized species in the magnetosphere.     

Comparison and analysis on lunar rotation with lunar gravity field models

Understanding the structure of and dynamic processes in the deep interior of planets is crucial for understanding their origin and evolution. An effective way to constrain them is through observation of rotation and subsequent simulation. In this paper, a numerical model of the Moon’s rotation and orbital motion is developed based on previous studies and implemented independently. The Moon is modeled as an anelastic body with a liquid core. The equations of the rotation were nonlinear and the Euler angles are cross coupled. We solve them numerically via the Runge-Kutta-Fehlberg (RKF) and multi-steps Adams-Bashforth-Moulton (ABM) predictor-corrector numerical integration. We have found that adequate accuracy is maintained by taking twelve steps per day using eleventh differences in the integrating polynomial. The lunar orbital and rotational equations are strongly coupled, so we integrated the rotation and motion simultaneously. We refer to other planetary informations from the newest planetary and lunar ephemeris INPOP17a, which is reported had fitted the longest LLR (Lunar Laser Ranging) observation data. Using the model GL660B from GRAIL (Gravity Recovery and Interior Laboratory) mission, we firstly compare our numerical results with the INPOP17a to prove the reasonability of our model. After that we apply the lunar gravity model CEGM02 determined from Chang’E-1 mission and SGM100h from SELENE mission to our model, the difference between results from CEGM02 and GL660B are less than \(-0.20 \sim0.15\) arc-second, and \(-0.25 \sim0.20\) arc-second for GL660B and SGM100h. Compared to SGM100h, the results show that the low degree and order coefficients (less than 6 from this paper) of lunar gravity field were improved in CEGM02 as expected. It is the first time to demonstrate that these models can be applied to lunar rotation model. These results manifest that a development of the gravity field measure will help us to know the rotation motion more precisely.     

Revealing Mars’ deep interior: Future geodesy missions using radio links between landers, orbiters, and the Earth

In the frame of a comparison between Earth, Venus, and Mars, a vision on future geodesy missions to Mars is discussed with particular focus on furthering our understanding of the interior, rotation, and orientation of this terrestrial planet. We explain how radioscience instruments can be used to observe the rotation and orientation and therewith to study the deep interior of Mars and its global atmosphere dynamics. Transponders in X-band and Ka-band are proposed with radio links between a lander or a rover and an orbiter around Mars and/or directly to the Earth. The radio budget links are studied in the frame of possible mission constraints and simulations are performed, which show that important information on the interior of Mars can be obtained from the radioscience data. From the observation of Mars’ orientation in space and of tidal effects on a spacecraft orbiting around Mars we show that it is possible for instance to constrain the dimension and composition of the core, the percentage of light element within the core, and to determine the presence of a pressure-induced mineral-phase transition at the bottom of the mantle.     

A timewise kinematic method for satellite gradiometry: GOCE simulations

We have defined new algorithms for the data processing of a satellite geodesy mission with gradiometer (such as the next European mission GOCE) to extract the information on the gravity field coefficients with a realistic estimate of their accuracy. The large scale data processing can be managed by a multistage decomposition. First the spacecraft position is determined, i.e., a kinematic method is normally used. Second we use a new method to perform the necessary digital calibration of the gradiometer. Third we use a multiarc approach to separately solve for the global gravity field parameters. Fourth we use an approximate resonant decomposition, that is we partition in a new way the harmonic coefficients of the gravity field. Thus the normal system is reduced to blocks of manageable size without neglecting significant correlations. Still the normal system is badly conditioned because of the polar gaps in the spatial distribution of the data. We have shown that the principal components of the uncertainty correspond to harmonic anomalies with very small signal in the region where GOCE is flying; these uncertainties cannot be removed by any data processing method. This allows a complete simulation of the GOCE mission with affordable computer resources. We show that it is possible to solve for the harmonic coefficients up to degree 200–220 with signal to error ratio ≥1, taking into account systematic measurement errors. Errors in the spacecraft orbit, as expected from state of the art satellite navigation, do not degrade the solution. Gradiometer calibration is the main problem. By including a systematic error model, we have shown that the results are sensitive to spurious gradiometer signals at frequencies close to the lower limit of the measurement band. If these spurious effects grow as the inverse of the frequency, then the actual error is larger than the formal error only by a factor ≃2, that is the results are not compromised.     

Geopotential from space techniques

The monitoring of the perturbed motion of artificial satellites since the 1960's has provided a valuable resource for improved activities in geodesy. These observations and the improved geodetic techniques they fostered have provided an unparalleled means for studying both the gravity field and the Earth's shape. In this paper, we will review the various determinations of the Earth's gravity field produced at many research centers over the years from space techniques. The specific methods which have been used to measure our planet, such as satellite altimetry and satellite orbital perturbation analyses (over short and long periods of time), will be compared.     

The Non-Gravitational Perturbations impact on the BepiColombo Radio Science Experiment and the key rôle of the ISA accelerometer: direct solar radiation and albedo effects

The paper is focused on the estimate of the impact of the non-gravitational perturbations on the orbit of the Mercury Planetary Orbiter (MPO), one of the two spacecrafts that will be placed in orbit around the innermost planet of the solar system by the BepiColombo space mission. The key rôle of the Italian Spring Accelerometer (ISA), that has been selected by the European Space Agency (ESA) to fly on-board the MPO, is outlined. In the first part of the paper, through a numerical simulation and analysis we have estimated, over a time span of several years, the long-period behaviours of the disturbing accelerations produced by the incoming direct solar radiation pressure, and the indirect effects produced by Mercury’s albedo. The variations in the orbital parameters of the spacecraft, together with their spectral contents, have been estimated over the analysed period. The direct solar radiation pressure represents the strongest non-gravitational perturbation on the MPO in the very complex radiation environment of Mercury. The order-of-magnitude of this acceleration is quite high, about 10^?6 m/s², because of the proximity to the Sun and the large area-to-mass ratio of the spacecraft. In the second part of the paper, we concentrated upon the short-period effects of direct solar radiation pressure and Mercury’s albedo. In particular, the disturbing accelerations have been compared with the ISA measurement error and the advantages of an on-board accelerometer are highlighted with respect to the best modelling of the non-gravitational perturbations in the strong radiation environment of Mercury. The readings from ISA, with an intrinsic noise level of about $10^{-9}\,m/s^{2}/\sqrt{Hz}The paper is focused on the estimate of the impact of the non-gravitational perturbations on the orbit of the Mercury Planetary Orbiter (MPO), one of the two spacecrafts that will be placed in orbit around the innermost planet of the solar system by the BepiColombo space mission. The key r?le of the Italian Spring Accelerometer (ISA), that has been selected by the European Space Agency (ESA) to fly on-board the MPO, is outlined. In the first part of the paper, through a numerical simulation and analysis we have estimated, over a time span of several years, the long-period behaviours of the disturbing accelerations produced by the incoming direct solar radiation pressure, and the indirect effects produced by Mercury’s albedo. The variations in the orbital parameters of the spacecraft, together with their spectral contents, have been estimated over the analysed period. The direct solar radiation pressure represents the strongest non-gravitational perturbation on the MPO in the very complex radiation environment of Mercury. The order-of-magnitude of this acceleration is quite high, about 10⁻⁶ m/s², because of the proximity to the Sun and the large area-to-mass ratio of the spacecraft. In the second part of the paper, we concentrated upon the short-period effects of direct solar radiation pressure and Mercury’s albedo. In particular, the disturbing accelerations have been compared with the ISA measurement error and the advantages of an on-board accelerometer are highlighted with respect to the best modelling of the non-gravitational perturbations in the strong radiation environment of Mercury. The readings from ISA, with an intrinsic noise level of about in the frequency band of 3·10⁻⁵–10⁻¹ Hz, guarantees a very significant reduction of the non-gravitational accelerations impact on the space mission accuracy, especially of the dominant direct solar radiation pressure.     

环月飞行器精密定轨的模拟仿真

以中国正在实施的探月计划“嫦娥1号”工程为背景，分析了在中国联合S波段(USB)测控网和甚长基线射电干涉(VLBI)跟踪网的现有空间分布、观测精度水平下的环月飞行器精密定轨．采用的方法是模拟仿真计算，即首先模拟观测数据，然后在计入各误差源的影响后进行求解，并对解算结果进行比较．模拟仿真的工具是美国宇航局哥达德飞行中心的空间数据分析软件系统GEODYN．环月飞行的主要误差源是月球重力场，为此首先讨论了目前精度最高的月球重力场模型JGL165P1的(形式)误差．在模拟了测距、测速以及VLBI的时延、时延率数据后，计入月球重力场的误差进行精密轨道确定．定轨时采用了减缩动力学(reduced dynamic)方法，即选用合适的经验加速度参数吸收重力场误差对定轨的影响．结果表明对于一个不将月球重力场作为主要科学目标的探月计划(如“嫦娥1号”)，减缩动力学方法是一个简单、有效地提高环月飞行器定轨精度的方法．     

Simulations using terrestrial geological analogues to assess interpretability of potential geological features of the Hermean surface restituted by the STereo imaging Camera of the SIMBIOSYS package (BepiColombo mission)

The BepiColombo space mission is one of the European Space Agency's cornerstone projects; it is planned for launch in 2013 to study the planet Mercury. One of the imaging instruments of BepiColombo is a STereo Camera (STC), whose main scientific objective is the global stereo mapping of the entire surface of Mercury. STC will permit the generation of a Digital Terrain Model (DTM) of Mercury's surface, improving the interpretation of morphological features at different scales and clarifying the stratigraphic relationships between different geological units.To evaluate the effectiveness of the STC-derived DTM for geological purposes, a series of simulations has been performed to find out to what extent the errors expected in the DTM may prevent the correct classification and interpretation of geological features. To meet this objective, Earth analogues (a crater, a lava cone and an endogenous dome) of likely components of the Hermean surface, small enough to be near the detection limit of the STC, were selected and a photorealistic three-dimensional (3D) model of each feature was generated using Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) stereo images. Stereoscopic pairs of synthetic images of each feature were then generated from the 3D model at different locations along the BepiColombo orbit. For each stereo pair, the corresponding Hermean DTM was computed using image correlation and compared to the reference data to assess the loss of detail and interpretability. Results show that interpretation and quantitative analysis of simple craters morphologies and small volcanic features should be possible all along the periherm orbit arc. At the apoherm only the larger features can be unequivocally distinguished, but they will be reconstructed to a poor approximation.     

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.     

The MESSENGER mission to Mercury: scientific objectives and implementation

Mercury holds answers to several critical questions regarding the formation and evolution of the terrestrial planets. These questions include the origin of Mercury's anomalously high ratio of metal to silicate and its implications for planetary accretion processes, the nature of Mercury's geological evolution and interior cooling history, the mechanism of global magnetic field generation, the state of Mercury's core, and the processes controlling volatile species in Mercury's polar deposits, exosphere, and magnetosphere. The MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) mission has been designed to fly by and orbit Mercury to address all of these key questions. After launch by a Delta 2925H-9.5, two flybys of Venus, and two flybys of Mercury, orbit insertion is accomplished at the third Mercury encounter. The instrument payload includes a dual imaging system for wide and narrow fields-of-view, monochrome and color imaging, and stereo; X-ray and combined gamma-ray and neutron spectrometers for surface chemical mapping; a magnetometer; a laser altimeter; a combined ultraviolet–visible and visible-near-infrared spectrometer to survey both exospheric species and surface mineralogy; and an energetic particle and plasma spectrometer to sample charged species in the magnetosphere. During the flybys of Mercury, regions unexplored by Mariner 10 will be seen for the first time, and new data will be gathered on Mercury's exosphere, magnetosphere, and surface composition. During the orbital phase of the mission, one Earth year in duration, MESSENGER will complete global mapping and the detailed characterization of the exosphere, magnetosphere, surface, and interior.     

On a nonlinear and lorentz-invariant version of Newtonian gravitation — II

This paper gives a full nonlinear version of Newtonian gravity in which the gravitational energy acts as a source of the gravitational field. The generalized field equation for the scalar gravitational potential is solved for a spherically symmetric localized distribution of matter. It is shown that the perihelia of orbits of test particles in such a field precess steadily. The effect is, however, too small to account for the observed shift in the perihelion of planet Mercury. Further, the bending of light in this theory is zero. It is suggested that these inadequacies of the quasi-Newtonian framework call for more sophisticated approaches to gravity.     

A procedure for determining the nature of Mercury's core

Abstract— We review the assertion that the precise measurement of the second degree gravitational harmonic coefficients, the obliquity, and the amplitude of the physical libration in longitude, C₂₀, C₂₂, θ, and φ₀, for Mercury are sufficient to determine whether or not Mercury has a molten core (Peale, 1976). The conditions for detecting the signature of the molten core are that such a core not follow the 88‐day physical libration of the mantle induced by periodic solar torques, but that it does follow the 250 000‐year precession of the spin axis that tracks the orbit precession within a Cassini spin state. These conditions are easily satisfied if the coupling between the liquid core and solid mantle is viscous in nature. The alternative coupling mechanisms of pressure forces on irregularities in the core‐mantle boundary (CMB), gravitational torques between an axially asymmetric mantle and an assumed axially asymmetric solid inner core, and magnetic coupling between the conducting molten core and a conducting layer in the mantle at the CMB are shown for a reasonable range of assumptions not to frustrate the first condition while making the second condition more secure. Simulations have shown that the combination of spacecraft tracking and laser altimetry during the planned MESSENGER (MErcury Surface, Space ENvironment, GEochemistry, Ranging) orbiter mission to Mercury will determine C₂₀, C₂₂, and θ to better than 1% and φ₀ to better than 8%—sufficient precision to distinguish a molten core and constrain its size. The possible determination of the latter two parameters to 1% or less with Earth‐based radar experiments and MESSENGER determination of C₂₀ and C₂₂ to 0.1% would lead to a maximum uncertainty in the ratio of the moment of inertia of the mantle to that of the whole planet, C_m/C, of ?2% with comparable precision in characterizing the extent of the molten core.     

Stereo topographic models of Mercury after three MESSENGER flybys

From photogrammetric analysis of stereo images of Mercury obtained during three MESSENGER flybys, we have produced three digital terrain models (DTMs) that have a grid spacing of 1 km and together cover 30% of the planet's surface. The terrain models provide a rich source of information on the morphology of Mercury's surface, including details of tectonic scarp systems as well as impact craters and basins. More than 400 craters larger than 15 km in diameter are included in the models. Additionally, the models provide important test cases for the analysis of stereo image data to be collected during MESSENGER's orbital mission phase. Small lateral offsets and differences in trends between stereo DTMs and laser altimeter profiles may be due to remaining errors in spacecraft position, instrument pointing, or Mercury coordinate knowledge. Such errors should be resolved during the orbital mission phase, when more joint analyses of data and detailed orbit modeling will be possible.