Accurate Mars Express orbits to improve the determination of the mass and ephemeris of the Martian moons

The determination of the ephemeris of the Martian moons has benefited from observations of their plane-of-sky positions derived from images taken by cameras onboard spacecraft orbiting Mars. Images obtained by the Super Resolution Camera (SRC) onboard Mars Express (MEX) have been used to derive moon positions relative to Mars on the basis of a fit of a complete dynamical model of their motion around Mars. Since, these positions are computed from the relative position of the spacecraft when the images are taken, those positions need to be known as accurately as possible. An accurate MEX orbit is obtained by fitting two years of tracking data of the Mars Express Radio Science (MaRS) experiment onboard MEX. The average accuracy of the orbits has been estimated to be around 20–25 m. From these orbits, we have re-derived the positions of Phobos and Deimos at the epoch of the SRC observations and compared them with the positions derived by using the MEX orbits provided by the ESOC navigation team. After fit of the orbital model of Phobos and Deimos, the gain in precision in the Phobos position is roughly 30 m, corresponding to the estimated gain of accuracy of the MEX orbits. A new solution of the GM of the Martian moons has also been obtained from the accurate MEX orbits, which is consistent with previous solutions and, for Phobos, is more precise than the solution from the Mars Global Surveyor (MGS) and Mars Odyssey (ODY) tracking data. It will be further improved with data from MEX-Phobos closer encounters (at a distance less than 300 km). This study also demonstrates the advantage of combining observations of the moon positions from a spacecraft and from the Earth to assess the real accuracy of the spacecraft orbit. In turn, the natural satellite ephemerides can be improved and participate to a better knowledge of the origin and evolution of the Martian moons.     

The theory of asynchronous relative motion I: time transformations and nonlinear corrections

Using alternative independent variables in lieu of time has important advantages when propagating the partial derivatives of the trajectory. This paper focuses on spacecraft relative motion, but the concepts presented here can be extended to any problem involving the variational equations of orbital motion. A usual approach for modeling the relative dynamics is to evaluate how the reference orbit changes when modifying the initial conditions slightly. But when the time is a mere dependent variable, changes in the initial conditions will result in changes in time as well: a time delay between the reference and the neighbor solution will appear. The theory of asynchronous relative motion shows how the time delay can be corrected to recover the physical sense of the solution and, more importantly, how this correction can be used to improve significantly the accuracy of the linear solutions to relative motion found in the literature. As an example, an improved version of the Clohessy-Wiltshire (CW) solution is presented explicitly. The correcting terms are extremely compact, and the solution proves more accurate than the second and even third order CW equations for long propagations. The application to the elliptic case is also discussed. The theory is not restricted to Keplerian orbits, as it holds under any perturbation. To prove this statement, two examples of realistic trajectories are presented: a pair of spacecraft orbiting the Earth and perturbed by a realistic force model; and two probes describing a quasi-periodic orbit in the Jupiter-Europa system subject to third-body perturbations. The numerical examples show that the new theory yields reductions in the propagation error of several orders of magnitude, both in position and velocity, when compared to the linear approach.     

Determination of the orbit of a natural satellite with hyperbolic flybys

We show that, when a natural satellite like Titan is invisible (e.g., due to an opaque atmosphere) its planetary orbit and its mass can be determined by tracking a spacecraft in close flybys. This is an important problem in the Cassini mission to the Saturnian system, which will be greatly improved by a good astrometric model for all its main components; in particular, an accuracy of a few hundred meters for the orbit of Titan is necessary to allow a measurement of its moment of inertia. The orbit of the spacecraft is the union of elliptical arcs, joined by short hyperbolic transitions: a problem of singular perturbation theory, whose solution leads to a matching condition between the inner hyperbolic orbit and the elliptical orbital elements. Since the inner elements are given in terms of the relative position and velocity of the spacecraft, accurate Doppler measurements in both regions can provide a satisfactory determination of Titan's position and velocity, hence of its Keplerian elements. The errors in this determination are discussed on the basis of the expected Allan deviation of the Doppler method; it is found that the driving errors are those in the elliptical arcs; the fractional errors in Titan's orbital elements are expected to be 10^–7. It is also possible to measure the mass of the satellite; however, when the eccentricity e of the flybys is large, the mass and a scaling transformation are highly correlated and the fractional error in the mass is expected to be e times worse.     

Invariant shape solutions of the spinning three craft Coulomb tether problem

In this paper we study shape-preserving formations of three spacecraft, where the formation keeping forces arise from the electric charges deposed on each craft. Inspired by Lagrange’s 3-body problem, the general conditions that guarantee preservation of the geometric shape of the electrically charged formation are derived. While the classical collinear configuration is a solution to the problem, the equilateral triangle configuration is found to only occur with unbounded relative motion. The three collinear spacecraft problem is analyzed and the possible solutions are categorized based on the spacecraft mass–charge ratio. Precise statements on the number of solutions associated with each category are provided. Finally, a methodology is proposed to study boundedness of the collinear solution that is inspired by past understanding and results for the 3-body problem. Given the initial position and the velocity vectors of each craft along with the charges, analytical solutions are provided describing the resulting relative motion.     

脉冲星在空间飞行器定位中的应用

利用脉冲星钟模型能高精度地预报脉冲星脉冲到达太阳系质心的时间。基于脉冲星时、空参考架可实现各类空间飞行器的自主导航。讨论了脉冲星钟的模型和脉冲星导航系统的框架结构,描述了脉冲星导航的基本原理和算法。指出脉冲星导航系统对脉冲星脉冲到达探测器时刻的测量精度,是决定空间飞行器位置解算精度的关键因素。脉冲星导航观测采用的原子钟如果足够稳定,则空间飞行器位置的解算方法可以简化。在脉冲星导航系统计时观测精度达到或优于几十微秒量级时,脉冲星视差、相对论效应的影响是不可忽略的。对脉冲星导航系统开发设计中的关键技术和进一步研究的主要问题进行了初步分析和讨论。     

On the effect of line current width and relative position on the multi-spacecraft curlometer technique

The response of the multi-spacecraft curlometer technique to variations in the size and relative position of infinitely long line currents with radially varying current density is systematically investigated for spacecraft in a regular tetrahedral formation. It is shown that, for line currents with a width less than the spacecraft separation, there is significant variation in the returned current with position of that current within the tetrahedron. For infinitely thin line currents, the curlometer tends to detect approximately 20% of the input current. For increasingly wide line currents there is less variation of the curlometer results with position of the current and the percentage of current magnitude detected increases. When the width of the current system is half the spacecraft separation, the curlometer tends to detect approximately 80% of the input current. These results are discussed in the context of multi-scale, multi-spacecraft missions.     

Orbit period modulation for relative motion using continuous low thrust in the two-body and restricted three-body problems

This paper presents rich new families of relative orbits for spacecraft formation flight generated through the application of continuous thrust with only minimal intervention into the dynamics of the problem. Such simplicity facilitates implementation for small, low-cost spacecraft with only position state feedback, and yet permits interesting and novel relative orbits in both two- and three-body systems with potential future applications in space-based interferometry, hyperspectral sensing, and on-orbit inspection. Position feedback is used to modify the natural frequencies of the linearised relative dynamics through direct manipulation of the system eigenvalues, producing new families of stable relative orbits. Specifically, in the Hill–Clohessy–Wiltshire frame, simple adaptations of the linearised dynamics are used to produce a circular relative orbit, frequency-modulated out-of-plane motion, and a novel doubly periodic cylindrical relative trajectory for the purposes of on-orbit inspection. Within the circular restricted three-body problem, a similar minimal approach with position feedback is used to generate new families of stable, frequency-modulated relative orbits in the vicinity of a Lagrange point, culminating in the derivation of the gain requirements for synchronisation of the in-plane and out-of-plane frequencies to yield a singly periodic tilted elliptical relative orbit with potential use as a Lunar far-side communications relay. The \(\Delta v\) requirements for the cylindrical relative orbit and singly periodic Lagrange point orbit are analysed, and it is shown that these requirements are modest and feasible for existing low-thrust propulsion technology.     

用脉冲星钟作航天器时间标准

在介绍参考坐标系和时间标准的基础上,讨论了用脉冲星为航天器导航的时间标准问题。利用X射线脉冲星实现航天器自主导航,星载钟的任何误差都会直接影响航天器位置测量。脉冲星钟具有较高的长期频率稳定度,适合用作各类航天器的时间标准。重点讨论了时间标准误差对航天器定位的影响;给出了用脉冲星钟作航天器时间标准的物理实现方法。     

"Quasi Satellite Orbits" to observe a possible small moon of Pallas

The purpose of this paper is to make a numerical search for natural orbits that can be used for a spacecraft to study a possible small moon of Pallas. There are many speculations about the existence of a small companion around this large asteroid, so finding and classifying orbits around this possible celestial body is an interesting problem in astrodynamics and that can be used for a spacecraft to observe this body. It is assumed that this moon has a radius that can vary from 0.125 to 1 km and that is located 750 or 500 km away from the center of Pallas. The idea is to show the effects of this parameter in the orbits around this moon. It means that the moon is much smaller than Pallas, so Keplerian orbits are not possible around it. To solve this problem, it is possible to find some special orbits that are called "Quasi Satellite Orbits" (QSO). They are orbits dominated by the gravity of Pallas, but that use the smaller perturbation from the moon to keep the spacecraft close to it. The present work searches for orbits that make the spacecraft to remain at given limits in its distance from the moon, like in the range from 3 to 50 km, the values used as an example in the present paper. This value is used because it is a good range to observe the body without getting to close to it, so reducing the risks of collisions. Each trajectory can be identified by the initial conditions of the spacecraft with respect to the moon, which means its initial position and velocity. The dynamics considers the restricted three-body problem and the influence of the solar radiation pressure, because some spacecraft may have higher values for the area-to-mass ratio, which gives a non-negligible effect in the trajectory of the spacecraft.     

The Solar Mass Ejection Imager (Smei)

We describe an instrument (SMEI) which has been specifically designed to detect and forecast the arrival of solar mass ejections and other heliospheric structures which are moving towards the Earth. Such events may cause geomagnetic storms, with resulting radiation hazards and disruption to military and commercial communications; damage to Earth-orbiting spacecraft; and also terrestrial effects such as surges in transcontinental power transmission lines. The detectors are sensitive over the optical wave-band, which is measured using CCD cameras. SMEI was launched on 6 January 2003 on the Coriolis spacecraft into a Sun-synchronous polar orbit as part of the US DoD Space Test Programme. The instrument contains three cameras, each with a field of view of 60°×3°, which are mounted onto the spacecraft such that they scan most of the sky every 102-min orbit. The sensitivity is such that changes in sky brightness equivalent to a tenth magnitude star in one square degree of sky may be detected. Each camera takes an image every 4 s. The normal telemetry rate is 128 kbits s^–1. In order to extract the emission from a typical large coronal mass ejection, stellar images and the signal from the zodiacal dust cloud must be subtracted. This requires accurate relative photometry to 0.1%. One consequence is that images of stars and the zodiacal cloud will be measured to this photometric accuracy once per orbit. This will enable studies of transient zodiacal cloud phenomena, flare stars, supernovae, comets, and other varying point-like objects.     

Algorithm Analysis of Autonomous Navigation of Spacecraft Based on X-ray Pulsars

A pulsar has the very stable rotation and can be used as the time standard. The astrometric parameters and astrophysical parameters of many pulsars, such as the spatial position, proper motion, distance, rotation period and its derivative, etc., can be all accurately determined. Since the pulsar can provide the time signal and the coordinates of its spatial position simultaneously, the pulsar navigation system installed on a spacecraft enables the autonomous navigation of the spacecraft to be realized. Firstly, the position of the spacecraft is predicted based on the equation of orbit dynamics of the spacecraft and then the Kalman filtering is applied to calculating the estimation error of the spacecraft position through the difference between the pulse arrival time observed on the spacecraft and the predicted pulse arrival time, thereby modifying the position of the spacecraft. Finally, the effects of the initial error, measuring accuracy of the pulse arrival time and number of pulsars on the navigation accuracy are analyzed.     

基于X射线脉冲星的航天器自主导航算法分析

脉冲星自转非常稳定,可以用作时间标准,许多脉冲星的空间位置、自行、距离、自转周期及其导数等天体测量参数和天体物理参数都能被精确测定.由于脉冲星能够同时提供时间信号和空间位置坐标,安装在航天器上的脉冲星导航系统能够实现航天器的自主导航.首先根据航天器轨道动力学方程预测航天器的位置,再通过航天器上观测的脉冲到达时间和预报的脉冲到达时间之差,应用Kalman滤波计算航天器位置估计的误差,从而对航天器的位置进行修正.最后,分析初始误差、脉冲到达时间测量精度、脉冲星个数对导航精度的影响.     

Calibrating the Pointing and Optical Parameters of the STEREO Heliospheric Imagers

The Heliospheric Imager (HI) instruments on the Solar TErrestrial RElations Observatory (STEREO) observe solar plasma as it streams out from the Sun and into the heliosphere. The telescopes point off-limb (from about 4° to 90° elongation) and so the Sun is not in the field of view. Hence, the Sun cannot be used to confirm the instrument pointing. Until now, the pointing of the instruments have been calculated using the nominal preflight instrument offsets from the STEREO spacecraft together with the spacecraft attitude data. This paper develops a new method for deriving the instrument pointing solutions, along with other optical parameters, by comparing the locations of stars identified in each HI image with the known star positions predicted from a star catalogue. The pointing and optical parameters are varied in an autonomous manner to minimise the discrepancy between the predicted and observed positions of the stars. This method is applied to all HI observations from the beginning of the mission to the end of April 2008. For the vast majority of images a good attitude solution has been obtained with a mean-squared deviation between the observed and predicted star positions of one image pixel or less. Updated values have been obtained for the instrument offsets relative to the spacecraft, and for the optical parameters of the HI cameras. With this method the HI images can be considered as “self-calibrating,” with the actual instrument offsets calculated as a byproduct. The updated pointing results and their by-products have been implemented in SolarSoft.     

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.     

Bounded relative motions for formation flying in displaced orbits by low-thrust propulsion

This paper discusses a numerical searching approach for the relative motion of formation flying in displaced orbits by spacecraft with low-thrust propulsion. The nonlinear dynamical model of spacecraft is established in a two-body rotating reference frame with arbitrary polar component of momentum and thrust-induced acceleration. Motions near the stable equilibria are distinguished from each other by means of five-dimensional variables, which can then be compressed uniquely into two-dimensional mapping images characterized by the crossing interval and the angle drifts. The surjective but not injective mapping makes the generation of three configurations of the relative motions possible. The corresponding relative orbits for three kinds of two-spacecraft formation flying are searched and exemplified based on the formation conditions formulized as functions of the crossing interval and the angle drifts. Furthermore, based on the assignment of displaced relative orbits to five-dimensional vector, the working orbit of the deputy for a specific chief can also be searched via the optimization algorithm to generate the bounded relative motion with the minimum thrust acceleration magnitude, which is of certain significance in reducing fuel consumption of formations.     

On one technique for determining the spacecraft attitude prom the sun

A variant to implement a spacecraft (SC) spatial attitude system with respect to the Sun is discussed. The sunward direction and the solar rotation axis are used as reference points. The system is based on measuring spectral line Doppler shift by scanning the solar image along the limb and is self-adjusting for relative spectral line shifts and instrument band shifts. The first harmonic of the signal serves as a basis for accurate adjustment of filter band. The second harmonic phase is used to measure the spacecraft attitude. The application of this method holds the greatest promise when the SO is stabilized by the sunward spinning because this ensures continuous monitoring of the spacecraft attitude.In addition, the method provides information on the precise coordinates of solar surface details during space-borne observations.     

A new method for resolving phase ambiguity in radio interferometry using Earth rotation synthesis

As a key technique in deep space navigation, radio interferometry can be used to determine the accurate location of a spacecraft in the plane-of-sky by measuring its signal propagation time delay between two remote stations. To improve the measurement accuracy, differential phase delay without phase ambiguity is usually desired. Aiming at the difficulties of resolving phase ambiguity with few stations and narrowband downlink signals, a new method is proposed in this work by taking advantage of the Earth rotation. The high accurate differential phase delay between the spacecraft and a calibrator can be achieved not only in the in-beam observation mode but also in the out-of-beam observation mode. In this paper we firstly built the model of phase ambiguity resolution. Then, main measurement errors of the model are analyzed, which is followed by tests and validations of the model and method using the tracking data of the Cassini mission and Chang'E-3 mission. The results show that the phase ambiguities can be correctly resolved to generate a 10-picosecond level accuracy differential phase delay. Angular measurement accuracy of the Cassini reaches the milli-arc-second level, and the relative position accuracy between the Chang'E-3 rover and lander reaches the meter level.     

Spacecraft Orbit Determination with The B-spline Approximation Method

It is known that the dynamical orbit determination is the most common way to get the precise orbits of spacecraft. However, it is hard to build up the precise dynamical model of spacecraft sometimes. In order to solve this problem, the technique of the orbit determination with the B-spline approximation method based on the theory of function approximation is presented in this article. In order to verify the effectiveness of this method, simulative orbit determinations in the cases of LEO (Low Earth Orbit), MEO (Medium Earth Orbit), and HEO (Highly Eccentric Orbit) satellites are performed, and it is shown that this method has a reliable accuracy and stable solution. The approach can be performed in both the conventional celestial coordinate system and the conventional terrestrial coordinate system. The spacecraft's position and velocity can be calculated directly with the B-spline approximation method, it needs not to integrate the dynamical equations, nor to calculate the state transfer matrix, thus the burden of calculations in the orbit determination is reduced substantially relative to the dynamical orbit determination method. The technique not only has a certain theoretical significance, but also can serve as a conventional algorithm in the spacecraft orbit determination.     

Astrodynamical Space Test of Relativity using Optical Devices I (ASTROD I)—a class-M fundamental physics mission proposal for cosmic vision 2015–2025: 2010 Update

ASTROD I is a planned interplanetary space mission with multiple goals. The primary aims are: to test General Relativity with an improvement in sensitivity of over 3 orders of magnitude, improving our understanding of gravity and aiding the development of a new quantum gravity theory; to measure key solar system parameters with increased accuracy, advancing solar physics and our knowledge of the solar system; and to measure the time rate of change of the gravitational constant with an order of magnitude improvement and the anomalous Pioneer acceleration, thereby probing dark matter and dark energy gravitationally. It is envisaged as the first in a series of ASTROD missions. ASTROD I will consist of one spacecraft carrying a telescope, four lasers, two event timers and a clock. Two-way, two-wavelength laser pulse ranging will be used between the spacecraft in a solar orbit and deep space laser stations on Earth, to achieve the ASTROD I goals.For this mission, accurate pulse timing with an ultra-stable clock, and a drag-free spacecraft with reliable inertial sensor are required. T2L2 has demonstrated the required accurate pulse timing; rubidium clock on board Galileo has mostly demonstrated the required clock stability; the accelerometer on board GOCE has paved the way for achieving the reliable inertial sensor; the demonstration of LISA Pathfinder will provide an excellent platform for the implementation of the ASTROD I drag-free spacecraft. These European activities comprise the pillars for building up the mission and make the technologies needed ready. A second mission, ASTROD or ASTROD-GW (depending on the results of ASTROD I), is envisaged as a three-spacecraft mission which, in the case of ASTROD, would test General Relativity to one part per billion, enable detection of solar g-modes, measure the solar Lense-Thirring effect to 10 parts per million, and probe gravitational waves at frequencies below the LISA bandwidth, or in the case of ASTROD-GW, would be dedicated to probe gravitational waves at frequencies below the LISA bandwidth to 100?nHz and to detect solar g-mode oscillations. In the third phase (Super-ASTROD), larger orbits could be implemented to map the outer solar system and to probe primordial gravitational-waves at frequencies below the ASTROD bandwidth. This paper on ASTROD I is based on our 2010 proposal submitted for the ESA call for class-M mission proposals, and is a sequel and an update to our previous paper (Appouchaux et al., Exp Astron 23:491?C527, 2009; designated as Paper I) which was based on our last proposal submitted for the 2007 ESA call. In this paper, we present our orbit selection with one Venus swing-by together with orbit simulation. In Paper I, our orbit choice is with two Venus swing-bys. The present choice takes shorter time (about 250?days) to reach the opposite side of the Sun. We also present a preliminary design of the optical bench, and elaborate on the solar physics goals with the radiation monitor payload. We discuss telescope size, trade-offs of drag-free sensitivities, thermal issues and present an outlook.     

Effects of nonthermal ions of distinct temperatures on dust acoustic shock waves in a dusty plasma

The nonlinear propagation of dust acoustic (DA) waves in an unmagnetized dusty plasma system consisting of negatively charged mobile dust fluid, Boltzmann distributed electrons, and two-temperature nonthermally distributed ions, is rigorously investigated. The reductive perturbation method has been employed to derive the Burgers equation. The hydrodynamic equation for inertial dust grains has been used to derive the Burgers equation. The effects of two temperature nonthermally distributed ions and dust kinematic viscosity, which are found to significantly modify the basic features of DA shock waves, are briefly discussed. Our present investigation can be effectively utilized in many astrophysical situations (e.g. satellite or spacecraft observations, Saturn’s E ring, etc.), which are discussed briefly in this analysis.