Dynamic orbit propagation in a gravitational field of an inhomogeneous attractive body using the Lagrange coefficients

The analytical methods have nearly been replaced by the numerical methods due to their higher accuracy and accessibility of computation facilities. The semi-analytical Lagrange method of orbit propagation using f and g series is a competitive alternative to the numerical integration technique if the Lagrange coefficients are derived in a full gravitational field. In this paper, a generalization of the Lagrange method of orbit propagation is introduced. In other words, we introduce a complete form of the Lagrange coefficients in all force fields developed in the spherical harmonics for example full gravitational field of the Earth. The method is numerically compared with the numerical integration technique. In order to show the numerical performance of the method, it has been implemented for orbit propagation of a GPS-like MEO and CHAMP-like LEO satellites. Discrepancy at centimeter level for CHAMP-like and sub-millimeter accuracy for GPS-like satellites shows relatively high performance of the developed algorithm. Compared to integration method, the proposed Lagrange method is nearly faster by a factor two for small Nmax and four for large Nmax.     

Detecting gravitational waves from accreting neutron stars

The gravitational waves emitted by neutron stars carry unique information about their structure and composition. Direct detection of these gravitational waves, however, is a formidable technical challenge. In a recent study we quantified the hurdles facing searches for gravitational waves from the known accreting neutron stars, given the level of uncertainty that exists regarding spin and orbital parameters. In this paper we reflect on our conclusions, and issue an open challenge to the theoretical community to consider how searches should be designed to yield the most astrophysically interesting upper limits. With this in mind we examine some more optimistic emission scenarios involving spin-down, and show that there are technically feasible searches, particularly for the accreting millisecond pulsars, that might place meaningful constraints on torque mechanisms. We finish with a brief discussion of prospects for indirect detection.     

Symplectic integration of space debris motion considering several Earth’s shadowing models

In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones. Indeed, it has already been demonstrated that such objects could stay there for hundreds of years. Our model takes into account the Earth’s gravitational potential, luni-solar and planetary gravitational perturbations and direct solar radiation pressure. Based on the analysis of the energy conservation and on a comparison with a high order non-symplectic integrator, we show that our algorithm allows us to use large time steps and keep accurate results. We also propose an innovative method to model Earth’s shadow crossings by means of a smooth shadow function. In the particular framework of symplectic integration, such a function needs to be included analytically in the equations of motion in order to prevent numerical drifts of the energy. For the sake of completeness, both cylindrical shadows and penumbra transitions models are considered. We show that both models are not equivalent and that big discrepancies actually appear between associated orbits, especially for high area-to-mass ratios.     

Features of vestibuloocular reflex modulations induced by altered gravitational forces in tadpoles (Xenopus laevis).

In Xenopus laevis tadpoles, we studied the static vestibuloocular reflex (rVOR) in relation to modifications of the gravitational environment to find basic mechanisms of how altered gravitational forces (AGF) affect this reflex. Animals were exposed to microgravity during space flight or hypergravity (3g) for 4 to 12 days. Basic observations were that (1)the development of the rVOR is significantly affected by altered gravitational conditions, (2) the duration of 1g-readaptation depends on the strength of the test stimulus, (3) microgravity induces malformations of the body which are related to the rVOR depression. Future studies are based on the hypotheses (1) that the vestibular nuclei play a key roll in the adaptation to AGF conditions, (2) that the stimulus transducing systems in the sense organ are affected by AGF conditions, and (3) that fertilized eggs will be converted to normal adults guided by physiological and morphological set points representing the genetic programs. Developmental retardation or acceleration, or otherwise occurring deviations from standard development during embryonic and postembryonic life will activate genes that direct the developmental processes towards normality.     

Modeling irregular small bodies gravity field via extreme learning machines and Bayesian optimization

Close proximity operations around small bodies are extremely challenging due to their uncertain dynamical environment. Autonomous guidance and navigation around small bodies require fast and accurate modeling of the gravitational field for potential on-board computation. In this paper, we investigate a model-based, data-driven approach to compute and predict the gravitational acceleration around irregular small bodies. More specifically, we employ Extreme Learning Machine (ELM) theories to design, train and validate Single-Layer Feedforward Networks (SLFN) capable of learning the relationship between the spacecraft position and the gravitational acceleration. ELM-base neural networks are trained without iterative tuning therefore dramatically reducing the training time. Analysis of performance in constant density models for asteroid 25143 Itokawa and comet 67/P Churyumov-Gerasimenko show that ELM-based SLFN are able learn the desired functional relationship both globally and in selected localized areas near the surface. The latter results in a robust neural algorithm for on-board, real-time calculation of the gravity field needed for guidance and control in close-proximity operations near the asteroid surface.     

The state of gravity sensors and peculiarities of plant growth during different gravitational loads.

In the primary roots of lettuce shoots grown under altered gravitational conditions--180 degrees inversion on the centrifuged clinostat, horizontal clinostat and in dynamic weightlessness--localization of the cellular organelles, cell morphology and peculiarities of growth have been studied. Significant changes took place in the localization of amyloplasts on the horizontal clinostat. The changes of amyloplast position in the cap cells on the horizontal clinostat and under weightlessness are similar. A change of the normal shoot position (180 degrees inversion and horizontal clinostat) causes an inhibition of growth. Weightlessness increases the length of axial organs and cells in the zone of elongation, but decreases the nitotic index in comparison to the centrifuged control. The anlysis of the formation of generative organs has been carried out for Arabidopsis plants grown on board the orbital station Salyut-6. The ability of plants to undergo vegetative growth and to pass through early phases of generative development under weightlessness was confirmed.     

The determination for ideal release point of test masses in drag-free satellites for the detection of gravitational waves

Gravitational waves are ripples in space–time predicted by Albert Einstein's general relativity and provide a new way to understand the universe. Space-borne detectors of gravitational waves, extending to very large scales, can effectively detect the middle and low-frequency gravitational wave source with the frequency band of 0.1 mHz–1 Hz. The test masses are used to make an inertial reference point in the detection of gravitational waves. Currently, there are few studies concerning the ideal release position for the test masses in the detection of gravitational waves. In this study, we give a general solution for test mass release points to minimize the relative motion between the test mass and the satellite mass center. Moreover, we discuss the situation when the release point equation is not satisfied, and the ideal release point of the along-track. Finally, we report on simulations that verify the accuracy of the theoretical derivation.     

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.     

双小行星系统引力场建模方法研究

针对弱引力双小行星系统的引力场建模问题,本文采用复杂度和精度依次递增的球体–球体模型、椭球体–球体模型和改进的限制性椭球体–椭球体模型来进行引力场建模,并分别采用椭圆积分以及无积分环节、计算效率高的二阶二次球谐函数来表征引力势,从而比较精确地刻画双小行星系统和探测器构成的限制性全三体问题的动力学模型;针对双小行星系统1999KW4,对其不同的引力场模型进行了仿真研究,分别给出了不同模型下的等效势能函数曲面及零速度曲线,比较了不同模型下的平动点位置坐标偏差。结果表明,二阶二次球谐函数计算引力势的椭球体-椭球体模型计算精度高,复杂程度低,计算量更少,计算速度更快,能够较精确的对双小行星系统进行引力场建模。     

Prediction of horizontal gas–solid flows under different gravitational fields

In this paper the performance of horizontal pneumatic conveying under different gravity environments is evaluated. An Euler–Lagrange approach validated versus ground experiments is employed to predict the relevant particle variables such as particle mass flux, mean conveying and fluctuating velocities in terrestrial, lunar and micro-gravity conditions. Gravity reduced computations predict a reduction in the global particle–wall collision frequency. Also, in the case of low wall roughness and small particle mass loading, reduction of gravity acceleration implies an increase of particle–wall collision frequency with the upper wall of the channel affecting greatly the particle mass flux profile. In the case of high wall roughness and/or high particle-to-fluid mass loading (i.e., around 1.0) particle conveying characteristics are similar in the three gravity conditions evaluated. This is due to the fact that both, wall roughness and inter-particle collisions reduce gravitational settling. However, the influence of gravity on the additional pressure loss along the channel due to the conveying of the particles is much reduced.     

基于最优控制的高超螺旋俯冲轨迹设计

针对高超声速飞行器在俯冲阶段的突防与制导问题，提出了一种基于最优控制的螺旋俯冲轨迹设计方法。首先，根据二阶视线角加速度相对运动方程强耦合、非线性的特点，引入了反馈线性化技术，将非线性系统转化成了线性系统；其次，为了让飞行器实现对目标的期望角度打击，在状态方程中引入了终端落角约束项。然后，以零化视线角速率和使消耗能量最小为目标，采用最优控制得到了加速度形式的控制输入。接着，为了让飞行器在空间内进行螺旋机动，设计了螺旋加速度控制飞行器速度矢量可绕瞬时视线轴进行旋转；最后，考虑到机动与制导的冲突，设计了高度函数使得最优与螺旋两种信号进行匹配。仿真结果显示，在面对相同的防空威胁时，与基于比例螺旋的轨迹相比，本文设计的俯冲轨迹具有更好的突防效果。     

Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence

This paper provides a hamiltonian formulation of the equations of motion of an artificial satellite or space debris orbiting the geostationary ring. This theory of order 1 has been formulated using canonical and non-singular elements for eccentricity and inclination. The analysis is based on an expansion in powers of the eccentricity and of the inclination. The theory accounts for the influence of the Earth gravity field expanded in spherical harmonics, paying a particular attention to the resonance occurring for geosynchronous objects. The luni-solar perturbations are also taken into account. We present the resonant motion and its main characteristics: equilibria, stability, fundamental frequencies and width of the resonant area by comparison with a basic analytical model. Finally, we show some results concerning the long term dynamics of a typical space debris under the influence of the gravitational field of the Earth and the luni-solar interactions.     

基于RCMAC网络的动态逆再入制导方法研究

针对升力式再入飞行器的制导问题,首先利用准平衡滑翔原理给出标准的阻力加速度．速度剖面,并对阻力加速度跟踪制导原理进行分析,然后利用自回归小脑模型神经网络（RCMAC）网络良好的非线性逼近能力、泛化能力和自学习能力,采用基于RCMAC网络的动态逆方法实现对阻力加速度的跟踪,并证明闭环系统的稳定性．三自由度仿真结果表明,该制导方式降低了动态逆方法对模型的依赖,增强了制导系统的鲁棒性．     

Studies of strong gravitational fields near super-massive black hole horizons with space missions

General relativity (GR) can be probed by several tests in the weak gravitational field limit. On the contrary, very poor information exists about GR tests in strong gravitational fields. Here, we focus on the interaction of light rays with the strong gravitational field of a massive black hole and show that relativistic images may form. Hence, we calculate the shapes of shadows (mirages) forming just near BH horizons and discuss the possibility to estimate the black hole parameters (mass, spin and charge) by future astrometric missions. In 2007, the Radioastron space telescope will be launched and it will allow to evaluate those parameters for the black hole hosted at the center of our Galaxy.     

Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery

The satellite gravity gradiometric data can be used directly to recover the gravity anomaly at sea level using inversion of integral formulas. This approach suffers by the spatial truncation errors of the integrals, but these errors can be reduced by modifying the formulas. It allows us to consider smaller coverage of the satellite data over the region of recovery. In this study, we consider the second-order radial derivative (SORD) of disturbing potential (T_rr) and determine the gravity anomaly with a resolution of 1° × 1° at sea level by inverting the statistically modified version of SORD of extended Stokes’ formula. Also we investigate the effect of the spatial truncation error on the quality of inversion considering noise of T_rr. The numerical investigations show satisfactory results when the area of T_rr coverage is the same with that of the gravity anomaly and the integral formula is modified by the biased least-squares modification. The error of recovery will be about 6 mGal after removing the regularization bias in the presence of 1 mE noise in T_rr measured on the orbit.     

An alternative computation of a gravity field model from GOCE

GOCE is the first satellite with a gravitational gradiometer (SGG). This allows to determine a gravity field model with high spatial resolution and high accuracy. Four of the six independent components of the gravitational gradient tensors (GGT) are measured with high accuracy in the so-called measurement band (MB) from 5 to 100 mHz by the GOCE gradiometer. Based on more than 1 year of GOCE measurements, two gravity field models have been derived. Here, we introduce a strategy for spherical harmonic analysis (SHA) from GOCE measurements, with a bandpass filter applied to the SGG data, combined with orbit analysis based on the integral equation approach, and additional constraints (or stabilization) in the polar areas where no observation is available due to the orbit geometry. In addition, we combined the GOCE SGG part with a set of GRACE normal equations. This improves the accuracy of the gravity field in the long-wavelength parts, due to the complementarity of GOCE and GRACE. Comparison with other models and with external data shows that our results are rather close to the GPS-levelling data in well-selected test regions, with an uncertainty of 4–7 cm, for truncation at degree 200.     

Evaluation of experiments involving the study of plant orientation and growth under different gravitational conditions.

The manifestation of gravitropic reaction in plants has been considered from the phylogenetic point of view. A chart has been suggested according to which it is supposed that the first indications of the ability to identify the direction of the gravitational vector were inherent in the most ancient eukaryotes, which gave rise to green, brown, yellow-green, golden and diatomaceous algae as well as fungi. The experiments on the role of gravity in plant ontogenesis are being continued. The sum total of the data obtained in a number of experiments in space shows that under these conditions a structurally modified but normally functioning gravireceptive apparatus is formed. The data confirming the modification, under changed gravity, of the processes of integral and cellullar growth of the axial organs of seedlings as well as of the anatomo-morphological structure and developmental rates of plants during their prolonged growth in space are presented. It is assumed that this fact testifies to the presence of systems interacting with gravity during plant ontogenesis. At the same time the necessity for further experiments in order to differentiate an immediate biological effect of gravity from the ones conditioned by it indirectly due to the changes in the behavior of liquids and gases is pointed out. The methodological aspects of biological experiments in space as the main source of reliable information on the biological role of gravity are discussed.     

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.     

Gravity field models from kinematic orbits of CHAMP,GRACE and GOCE satellites

The aim of our work is to generate Earth’s gravity field models from GPS positions of low Earth orbiters. Our inversion method is based on Newton’s second law, which relates the observed acceleration of the satellite with forces acting on it. The observed acceleration is obtained as numerical second derivative of kinematic positions. Observation equations are formulated using the gradient of the spherical harmonic expansion of the geopotential. Other forces are either modelled (lunisolar perturbations, tides) or provided by onboard measurements (nongravitational perturbations). From this linear regression model the geopotential harmonic coefficients are obtained.