Libration-Generated Average Convection in a Rotating Flat Layer with Horizontal Axis

The study of average convection in a rotating cavity subjected to modulated rotation is an interesting area for the development of both fundamental and applied science.This phenomenon finds application in the field of mass transfer and fluid flow control,relevant examples being crystal growth under reduced gravity and fluid mixing in microfluidic devices for cell cultures.In this study,the averaged flow generated by the oscillating motion of a fluid in a planar layer rotating about a horizontal axis is experimentally investigated.The boundaries of the layer are maintained at constant temperatures,while the lateral cylindrical wall is thermally insulated.It is demonstrated that libration results in intense oscillatory fluid motion,which in turn produces a time-averaged flow.For the first time,quantitative measures for the instantaneous velocity field are obtained using the Particle Image Velocimetry technique.It is revealed that the flow has the form of counter-rotating vortices.The vortex circulations sense changes during a libration cycle.An increase in the rotation rate and amplitude of the cavity libration results in an increase in the flow intensity.The heat transfer and time-averaged velocity are examined accordingly as a function of the dimensionless oscillation frequency,and resonant excitation of heat transfer and average oscillation velocity are revealed.The threshold curve for the onset of the averaged convection is identified in the plane of control parameters(dimensionless rotational velocity and pulsation Reynolds number).It is found that an increase in the dimensionless rotational velocity has a stabilizing effect on the onset of convection.

Determination of the free lunar libration modes from ephemeris DE430

The Moon’s physical librations have been extensively studied, and elaborate researches have been developed for the purpose of deriving accurate modes of free librations. Our motivation comes from the Planetary and Lunar Ephemeris DE430 by JPL/NASA, which was created in April 2013,and is reported to be the most accurate lunar ephemeris today using the data from Gravity Recovery and Interior Laboratory（GRAIL）. Therefore, the residuals after fitting the model have reduced owing to improvement in the libration models, and the free librations embedded in the Euler angles have also improved. We use Fourier analysis to extract the approximate frequencies from DE430 and then a quadratic interpolation method is used to determine higher accuracy frequencies. With the frequencies,the linear least-squares fitting method is employed to fit the lunar physical librations to DE430. From this analysis we identified the three modes of free physical librations, and estimated the amplitudes as 1.471′′in longitude, 0.025′′in latitude and 8.19′′× 3.31′′for the wobble, with the respective periods of1056.16, 8806.9 and 27262.99 d. Since the free librations damp with time, they require recent excitation or a continuous stimulating mechanism in order to sustain.

The Libration Points in Photogravitational Restricted Three-Body Problem

The phologravilational restricled three-body problem in which the mass reduclionfactors of two primaries q_1 q_2( -∞, 1] are siudied and an analytic meihod toesli,;iale the number of libralion points ana io calculate lheir hoalion is given in thispaper. The results show lhal in phologravilalional reslricled three-body problem, thenumber of librafion poinis is .from one to seven for different q_1 and q_2. As application,the motion of dust grain like comet tail in the solar syslern is also discussed.

Improved semi-analytical computation of center manifolds near collinear libration points

In the framework of the circular restricted three-body problem, the center manifolds associated with collinear libration points contain all the bounded orbits moving around these points. Semianalytical computation of the center manifolds and the associated canonical transformation are valuable tools for exploring the design space of libration point missions. This paper deals with the refinement of reduction to the center manifold procedure. In order to reduce the amount of calculation needed and avoid repetitive computation of the Poisson bracket, a modified method is presented. By using a polynomial optimization technique, the coordinate transformation is conducted more efficiently. In addition, an alternative way to do the canonical coordinate transformation is discussed, which complements the classical approach. Numerical simulation confirms that more accurate and efficient numerical exploration of the center manifold is made possible by using the refined method.

A new control law based on characteristic exponent assignment for libration point orbit maintenance

A new method is developed for stabilizing motion on collinear libration point orbits using the formalism of the circular restricted three body problem. Linearization about the collinear libration point orbits yields an unstable linear parameter-varying system with periodic coefficients. Given the variational equations, an innovative control law based on characteristic exponent assignment is introduced for libration point orbit maintenance. A numerical simulation choosing the Richardson＇s third order approximation for a halo orbit as a nominal orbit is conducted, and the results demonstrate the effectiveness of this control law.

A station-keeping strategy for collinear libration point orbits

Spacecrafts in periodic or quasi-periodic orbits near the collinear libration points are proved to be excellent platforms for scientific investigations of various phenomena.Since such periodic or quasi-periodic orbits are exponentially unstable,the station-keeping maneuver is needed. A station-keeping strategy which is found by an analytical method is presented to eradicate the dominant unstable component of the libration point trajectories.The inhibit force transforms the unstable component to a stable component.In this method,it is not necessary to determine a nominal orbit as a reference path.

Motion modeling and formation form analysis of spacecraft near collinear libration points

To meet the increasing research demand for deep space exploration,especially for the second libration point (L2) conditional periodic orbit (Halo orbit) in the Sun-Earth system,the methods to get analytical Halo orbit and differential-correction Halo orbit were described firstly,and the corresponding orbits accuracy was analyzed.Then,based on the results of third-order and differential-correction Halo orbits,the formation form was studied.Analysis was carried out to discuss the influence of system amplitude,initial phase,and phase difference on the formation form,as well as that of initial orbit values on form accuracy.Finally,some simulation results demonstrate the validity of the proposed methods.

Lissajous Orbits at the Collinear Libration Points in the Restricted Three-Body Problem with Oblateness

In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissajous orbits and the phase spaces are obtained under the effect of oblateness.

Libration Points in the R3BP with a Triaxial Rigid Body as the Smaller Primary and a Variable Mass Infinitesimal Bod

The paper deals with the existence of the coplanar libration points in the restricted three-body problem when the smaller primary is a triaxial rigid body and the infinitesimal body is of variable mass. Following small parameter method, the coordinates of collinear libration points are established whereas the coordinates of triangular libration points are established by classical method. It is found that the mass reduction factor has small effect but triaxiality parameters of the smaller primary have great effects on the coordinates of the libration points.

Multi-modes control method for spacecraft formation establishment and reconfiguration near the libration points

This paper studies Multi-modes control method for libration points formation establishment and reconfiguration. Firstly, relations between optimal impulse control and Floquet modes are investigated. Method of generating modes is proposed. Characteristics of the mode coefficients stimulated at different time are also given. Studies show that coefficients of controlled modes can be classified into four types, and formation establishment and reeonfiguration can be achieved by multi-impulse control with the presented method of generating modes. Then, since libration points formation is generally unstable, mutli-modes keeping control method which can stabilize five Floquet modes simultaneously is proposed. Finally, simulation on formation establishment and reconfiguration are carried out by using method of generating modes and mutli-modes keeping control method. Results show that the proposed control method is effective and practical.

Analytical libration control law for electrodynamic tether system with current constraint

This study focuses on stabilizing the libration dynamics of an electrodynamic tether system(EDTS)using generalized torques induced by the Lorentz force.In contrast to existing numerical optimization methods,a novel analytical feedback control law is developed to stabilize the in-plane and out-of-plane motions of a tether by modulating the electric current only.The saturation constraint on the current is accounted for by adding an auxiliary dynamic system to the EDTS.To enhance the robustness of the proposed controller,multiple perturbations of the orbital dynamics,modeling uncertainties,and external disturbances are approximated using a neural network in which the weighting matrix and approximation error are estimated simultaneously,such that these perturbations are well compensated for during the control design of the EDTS.Furthermore,a dynamically scaled generalized inverse is utilized to address the singular matrix in the control law.The closed-loop system is proven to be ultimately bounded based on Lyapunov stability theory.Finally,numerical simulations are performed to demonstrate the effectiveness of the proposed analytical control law.

Quasi-periodic orbits of small solar sails with time-varying attitude around Earth–Moon libration points

This paper proposes new quasi-periodic orbits around Earth–Moon collinear libration points using solar sails.By including the time-varying sail orientation in the linearized equations of motion for the circular restricted three-body problem(CR3BP),four types of quasi-periodic orbits(two types around L1 and two types around L2)were formulated.Among them,one type of orbit around L2 realizes a considerably small geometry variation while ensuring visibility from the Earth if(and only if)the sail acceleration due to solar radiation pressure is approximately of a certain magnitude,which is much smaller than that assumed in several previous studies.This means that only small solar sails can remain in the vicinity of L2 for a long time without propellant consumption.The orbits designed in the linearized CR3BP can be translated into nonlinear CR3BP and high-fidelity ephemeris models without losing geometrical characteristics.In this study,new quasi-periodic orbits are formulated,and their characteristics are discussed.Furthermore,their extendibility to higher-fidelity dynamic models was verified using numerical examples.

Autonomous navigation to quasi-periodic orbits near translunar libration points

Libration-point missions have been very useful and successful. Due to the unstable natures of most of these orbits, the long-time stationkeeping demands frequent maneuvers and precise orbit determinations. Earth-based tracking will have to undertake much more responsibilities with the increasing number of libration missions. An autonomous navigation system could offer a better way to decrease the need for Earth-based tracking. Nevertheless, when an autonomous navigation system is applied, there are three important factors affecting autonomous navigation accuracy, i.e., the accuracy of initial conditions, the accuracy of measurements, and the accuracy of onboard dynamics for propagation. This paper focuses on analyzing the influence from the third factor and finding an appropriate navigation dynamics, which can satisfy the requirement of estimation accuracy but not cause too much burden for onboard computation. When considering the restricted three-body model and the bicircular restricted four-body model as navigation dynamics, the astrin- gency is not shown during the simulations. Meanwhile, when considering the influences of the Sun＇s direct and indirect perturbations and the eccentricity of the Moon＇s orbit, a new navigation dynamic model with the standard ephemerides is proposed. The simulation shows the feasibility of the proposed model.

Trajectory design for the Moon departure libration point mission in full ephemeris model

The lunar probe may still have some remaining fuel after completing its predefined Moon exploration mission and is able to carry out some additional scientific or technological tasks after escaping from the Moon orbit.The Moon departure mission for the lunar probe is the focus of this paper.The possibility of the spacecraft orbiting the Moon to escape the Moon's gravitational pull is analyzed.The trajectory design for the Earth-Moon system libration point mission is studied in a full ephemeris dynamical model,which considers the non-uniform motion of the Moon around the Earth,the gravity of the Sun and planets and the finite thrust of the onboard engine.By applying the Particle Swarm Optimization algorithm,the trajectory design for the transfer from the Moon-centered orbit to the L1 halo orbit,the station-keeping strategies for the Earth-Moon halo orbit and the construction of homoclinic and heteroclinic orbits are investigated.Taking the tracking conditions and engineering constraints into account,two feasible schemes for the Moon departure libration point mission for the lunar probe are presented.

Libration and end body swing stabilization of a parallel partial space elevator system

This paper studies the libration and stabilization of a parallel partial space elevator system in circular orbits. The system is made up of two paralleled partial space elevators, each of which consists of one main satellite, one end body and a climber moving along the tether between them.The libration characteristics of the elevator are studied through numerical analysis by a new dynamic model, and a novel control strategy is proposed to stabilize the swing of the end body by projecting the climber speeds only. Optimal control method is used to implement the new control strategy in the case where the climbers move in opposite direction. The simulation results validate the effectiveness of the proposed control strategy whose application will neither sacrifice the transport efficiency nor exacerbate libration significantly.

Lunar project ILOM:application of the analytical theory of Lunar physical libration for the simulation of star observations

This study briefly describes the targets and problems of the future Japanese project In situ Lunar Orientation Measurement(ILOM),which is planned for the year 2017.One of the important parts of the project is to place a small optical telescope on the Lunar surface with the purpose to detect the Lunar physical libration with unprecedented accuracy 0.001 arcsec.At the present stage of research the computer simulation of future observations is going on,aiming to determine the moments of transition of a star through the first meridian and the polar distance of the star.Rotation of the Moon is being calculated under the analytical theory developed in the frame of a theme of the grant.A list of stars brighter than 12 m,whose coordinates are close to the Lunar precession pole motion,was constructed on the basis of several star catalogues.On average,for each moment of observation in the field of view of the telescope(1°) there are approximately 20-25 stars.Analyses of simulated stellar tracks observable from the Lunar surface(in a polar zone) reveal the significant difference from daily parallels of stars in comparison with the Earth.During one Lunar "day" equal 237 terrestrial days,a star moves on a spiral.However,depending on a longitude of a star,these spirals can be untwisted or twisted.In the latter case a star can describe a loop in the sky of the Moon during the period of supervision.Such an unusual astrometric phenomenon combined with the slow rotation of the Moon is compared with the Earth and the fast precession motion of the Lunar pole(in comparison with the precession motion of a terrestrial pole).

On-orbit calibration of soft X-ray detector on Chang'E-2 satellite

The X-ray spectrometer is one of the satellite payloads on the Chang＇E-2 satellite. The soft X-ray detector is one of the devices on the X-ray spectrometer, designed to detect the major rock-forming elements within the 0.5-10 keV range on the lunar surface. In this paper, energy linearity and energy resolution calibration is done using a weak 55Fe source. Temperature and time effects are found not to give a large error. The total uncertainty of calibration is estimated to be within 5% after correction.

A symplectic moving horizon estimation algorithm with its application to the Earth-Moon L2 libration point navigation

Accurate state estimations are perquisites of autonomous navigation and orbit maintenance missions.The extended Kalman lter(EKF)and the unscented Kalman lter(UKF),are the most commonly used method.However,the EKF results in poor estimation performance for systems are with high nonlinearity.As for the UKF,irregular sampling instants are required.In addition,both the EKF and the UKF cannot treat constraints.In this paper,a symplectic moving horizon estimation algorithm,where constraints can be considered,for nonlinear systems are developed.The estimation problem to be solved at each sampling instant is seen as a nonlinear constrained optimal control problem.The original nonlinear problem is transferred into a series of linear-quadratic problems and solved iteratively.A symplectic method based on the variational principle is proposed to solve such linear-quadratic problems,where the solution domain is divided into sub-intervals,and state,costate,and parametric variables are locally interpolated with linear approximation.The optimality conditions result in a linear complementarity problem which can be solved by the Lemke's method easily.The developed symplectic moving horizon estimation method is applied to the Earth-Moon L2 libration point navigation.And numerical simulations demonstrate that though more time-consuming,the proposed method results in better estimation performance than the EKF and the UKF.

Libration control of bare electrodynamic tether for three-dimensional deployment

Various promising applications of electrodynamic tether have been proposed for space missions over the past decades.A crucial issue of these missions is to deploy an electrodynamic tether under a rapid and stable state.This paper aims to stabilize the libration motions of a bare electrodynamic tether during its three-dimensional deployment.The tethered system under consideration consists of a main-satellite and a sub-satellite connected to each other through a bare electrodynamic tether.A widely used dumbbell assumption considering the tether as rigid and inflexible is adopted to facilitate the dynamic modeling and analysis of the tethered system.A pair of active control laws is synthesized by simultaneously regulating the electric current and tether tension to achieve an efficient stabilization of the three-dimensional libration of the bare electrodynamic tether in the deployment process.Moreover,comparisons of three groups of numerical simulations are performed to evaluate the influences of orbital inclinations and geomagnetic field models and the performance of the active control laws.

THE DETERMINATION OF THE MOON'S FREE LIBRATIONS BY LLR METHOD

Ⅰ. INTRODUCTIONBesides the theoretical meaning of determining the flee librations itself, we may also infer the excitation and dissipation mechanisms, such as meteorite impact, moonquakes and turbulent core-mantle frictions. Therefore, it will be of great significance to realize the origins, evolutions and structures of the bodies in the solar system.