Analytic and numerical treatment of motion of dust grain particle around triangular equilibrium points with post-AGB binary star and disc

This paper investigates the motion around the triangular equilibrium points, of a passively gravitating dust particle in the gravitational field of a low-mass post-AGB binary system, surrounded by circumbinary disc. The two bodies of the binary are modeled as a triaxial star and a radiating-oblate star. Due to small deviation of disc stars on circular orbits, we have assumed that the Coriolis and centrifugal forces of the stars are slightly perturbed. The triangular equilibrium points of the particle are found. These points are defined by, triaxiality of the primary star, oblateness and radiation of the secondary one and the gravitational potential from the disc mass. Further, when the disc mass increases, the particle moves nearer to the stars and farther away from the disc. In general, these equilibrium points are linearly stable when μ < μ_C; where μ is the mass ratio and μ_C is the critical mass function, defined by the parameters of the system. The effects of each of these parameters on the size of the stability region are stated, and the periodic motion around the stable points is examined. It is seen that the orbits are ellipses, and the orientation, eccentricities, lengths of the semi-major and semi-minor axes are influenced by the parameters of the problem. In particular, for our numerical linear stability analysis, we have taken an extremely depleted pulsating star, IRAS 11472-0800 as the post-AGB triaxial star, with a weakly-radiating young white dwarf star; G29-38 as the secondary. For this system, the stability result of the triangular points comes out different. Here, μ_C < μ throughout the entire range of the mass ratio and the critical mass function. Hence, the triangular equilibrium points are unstable. The stability of the orbits is tested using the Poincaré surfaces of section (Pss). The region of stability is controlled by the introduced parameters and the Jacobi constant.     

Families of halo orbits in the elliptic restricted three-body problem for a solar sail with reflectivity control devices

Solar sail halo orbits designed in the Sun-Earth circular restricted three-body problem (CR3BP) provide inefficient reference orbits for station-keeping since the disturbance due to the eccentricity of the Earth’s orbit has to be compensated for. This paper presents a strategy to compute families of halo orbits around the collinear artificial equilibrium points in the Sun-Earth elliptic restricted three-body problem (ER3BP) for a solar sail with reflectivity control devices (RCDs). In this non-autonomous model, periodic halo orbits only exist when their periods are equal to integer multiples of one year. Here multi-revolution halo orbits with periods equal to integer multiples of one year are constructed in the CR3BP and then used as seeds to numerically continue the halo orbits in the ER3BP. The linear stability of the orbits is analyzed which shows that the in-plane motion is unstable while the out-of-plane motion is neutrally stable and a bifurcation is identified. Finally, station-keeping is performed which shows that a reference orbit designed in the ER3BP is significantly more efficient than that designed in the CR3BP, while the addition of RCDs improve station-keeping performance and robustness to uncertainty in the sail lightness number.     

Dynamics of a spacecraft and normalization around Lagrangian points in the Neptune–Triton system

The problem of a spacecraft orbiting the Neptune–Triton system is presented. The new ingredients in this restricted three body problem are the Neptune oblateness and the high inclined and retrograde motion of Triton. First we present some interesting simulations showing the role played by the oblateness on a Neptune’s satellite, disturbed by Triton. We also give an extensive numerical exploration in the case when the spacecraft orbits Triton, considering Sun, Neptune and its planetary oblateness as disturbers. In the plane a × I (a = semi-major axis, I = inclination), we give a plot of the stable regions where the massless body can survive for thousand of years. Retrograde and direct orbits were considered and as usual, the region of stability is much more significant for the case of direct orbit of the spacecraft (Triton’s orbit is retrograde). Next we explore the dynamics in a vicinity of the Lagrangian points. The Birkhoff normalization is constructed around L₂, followed by its reduction to the center manifold. In this reduced dynamics, a convenient Poincaré section shows the interplay of the Lyapunov and halo periodic orbits, Lissajous and quasi-halo tori as well as the stable and unstable manifolds of the planar Lyapunov orbit. To show the effect of the oblateness, the planar Lyapunov family emanating from the Lagrangian points and three-dimensional halo orbits are obtained by the numerical continuation method.     

Configuration space and stability analysis of solar sail near-vertical earth-trailing orbits

Regions outside the reach of traditional propulsion systems or the ones that require significant propellant, may be reached by harnessing the solar radiation pressure and leveraging coupled dynamics to maneuver a sail-based spacecraft. Earth-trailing orbits have recently been investigated for getting a unique perspective of the Sun while maintaining the spacecraft in close proximity to Earth. Vertical orbits trailing the Earth exhibit the additional capability to view the Sun from above and below the ecliptic plane. In this work, families of sail-based orbits are explored for varying Earth-trailing angles and Z amplitudes in the Sun-Earth circular restricted three-body problem. Optimization is carried out to ensure that the non-traditional vertical orbits exhibit a constant pitch angle control history, as well as symmetry across the X-Y plane. The stability of the resulting orbit families is assessed using an extension of Flouquet theory to Differential Algebraic Equations. Results indicate that sail-based Earth-trailing vertical orbits can be more stable than traditional sub-L₁ sail-based vertical orbits.     

限制性三体问题共线平动点轨道近似解析解

随着深空探测成为航天领域的研究热点,与其密切相关的三体问题基础研究也日益重要,尤其是在深空探测任务设计中处于基础地位的共线平动点附近运动的研究,更是具有重要的工程应用价值。在圆型限制性三体问题下,对共线平动点附近运动近似解析解的研究已经较为全面,但在更接近真实情况、更具一般性的椭圆型限制性三体问题下,相应的研究却相对较少。针对此背景,参考借鉴圆型限制性三体问题的研究方法,首先根据平动点的特性计算出平动点的位置,然后将非线性三体动力学模型在共线平动点处线性化,最后结合线性系统理论,获得了椭圆型限制性三体问题下共线平动点附近运动的近似解析解,并将其与经典的圆型限制性三体问题下的近似解析解进行对比分析,仿真结果证明了方法的有效性,同时也表明所推导的椭圆型限制性三体问题解析解相比圆型限制性三体问题解析解具有更高的精度。     

Orbital stability close to asteroid 624 Hektor using the polyhedral model

We investigate the orbital stability close to the unique L4-point Jupiter binary Trojan asteroid 624 Hektor. The gravitational potential of 624 Hektor is calculated using the polyhedron model with observational data of 2038 faces and 1021 vertexes. Previous studies have presented three different density values for 624 Hektor. The equilibrium points in the gravitational potential of 624 Hektor with different density values have been studied in detail. There are five equilibrium points in the gravitational potential of 624 Hektor no matter the density value. The positions, Jacobian, eigenvalues, topological cases, stability, as well as the Hessian matrix of the equilibrium points are investigated. For the three different density values the number, topological cases, and the stability of the equilibrium points with different density values are the same. However, the positions of the equilibrium points vary with the density value of the asteroid 624 Hektor. The outer equilibrium points move away from the asteroid’s mass center when the density increases, and the inner equilibrium point moves close to the asteroid’s mass center when the density increases. There exist unstable periodic orbits near the surface of 624 Hektor. We calculated an orbit near the primary’s equatorial plane of this binary Trojan asteroid; the results indicate that the orbit remains stable after 28.8375?d.     

Low-energy near Earth asteroid capture using Earth flybys and aerobraking

Since the Sun-Earth libration points L₁ and L₂ are regarded as ideal locations for space science missions and candidate gateways for future crewed interplanetary missions, capturing near-Earth asteroids (NEAs) around the Sun-Earth L₁/L₂ points has generated significant interest. Therefore, this paper proposes the concept of coupling together a flyby of the Earth and then capturing small NEAs onto Sun–Earth L₁/L₂ periodic orbits. In this capture strategy, the Sun-Earth circular restricted three-body problem (CRTBP) is used to calculate target Lypaunov orbits and their invariant manifolds. A periapsis map is then employed to determine the required perigee of the Earth flyby. Moreover, depending on the perigee distance of the flyby, Earth flybys with and without aerobraking are investigated to design a transfer trajectory capturing a small NEA from its initial orbit to the stable manifolds associated with Sun-Earth L₁/L₂ periodic orbits. Finally, a global optimization is carried out, based on a detailed design procedure for NEA capture using an Earth flyby. Results show that the NEA capture strategies using an Earth flyby with and without aerobraking both have the potential to be of lower cost in terms of energy requirements than a direct NEA capture strategy without the Earth flyby. Moreover, NEA capture with an Earth flyby also has the potential for a shorter flight time compared to the NEA capture strategy without the Earth flyby.     

Intracellular molecular distributions in spacecraft experiments in orbit around Earth

It is possible that the nucleolous inside the cell plays the role of a “gravity receptor”. Furthermore, cells up to 10 μm in diameter can demonstrate some effect due to the redistribution of mitochondria or nucleolous. Effects of gravity should be present in various cell systems where larger objects such as the ribosomes move from cell to cell. In this paper we study the effects of gravity on cells. In particular, we examine the resulting intracellular molecular distribution due to Brownian motion and the ordered distribution of molecules under the action of gravity, where n₀ is the number per unit volume at certain level, and n is the number per unit volume above that level. This is an experiment that takes place at a certain orbital altitude in a spacecraft in orbit around Earth, where the acceleration due to the central field is corrected for the oblateness and also the rotation of the Earth. We found that equatorial circular and elliptical orbits have the highest n/n₀ ratios. This experiment takes place in circular and elliptical orbits, with eccentricities e = 0, 0.1 and involves a bacterial cell at an orbital altitude of 300 km. We found that n/n₀ = 1.00299 and 1.0037 respectively, which is still a 0.6–0.7 % higher than n/n₀ = 0.0996685 calculated on the surface of the Earth. Examining mitochondria in similar orbital experiments we found that equatorial orbits result to higher n/n₀ ratios. In particular, we found that n/n₀ = 8.38119, where an elliptical orbit of eccentricity e = 0.1 results to n/n₀ = 13.8525. Both are high above 100%, signifying the importance of Brownian motion over gravity. Our results are of interest to biomedical applications. Molecular concentrations are important for various processes such as the embryogenesis, positional homeostasis and its relation to cell energy expenditure, cell torque, cell deformation, and more. These results indicate that statistical molecular distributions play an important role for the recognition of a particular environment by the cell, in biological space experiment to come.     

Formation flying for magnetic sails around artificial equilibrium points

This paper investigates the problem of magnetic sail-based spacecraft formation control around the artificial equilibrium points (AEPs), which can eliminate the requirement of the propellant. The thrusts are achieved by utilizing the interaction between the solar wind and the artificial magnetosphere generated by superconducting current coil onboard. The circular restricted three-body problem (CRTBP) of magnetic sail is discussed including the allowed regions and linear stability of AEPs, the locations of collinear AEPs and the possibility of existence of periodic orbits around the collinear AEPs. Next, the dynamical models of magnetic sail formation around the collinear AEPs are established. A novel fast fixed-time nonsingular terminal sliding mode controller (FFNTSM) based on fixed-time disturbance observer (FTDO) is developed to account for external disturbances. Several numerical simulations are conducted to substantiate that spacecraft formation can be precisely controlled by the proposed propellantless propulsion method in the presence of external disturbances.     

Hierarchical global search for fuel-optimal impulsive transfers between lunar libration orbits

The attention to the periodic orbit in the Earth-Moon restricted three-body system continues to grow due to its special environment and locations. This research investigates the feasibility of constructing fuel-optimal single and multiple impulse transfers between unstable periodic orbits at L₁ and L₂ points. Invariant manifolds, which could provide the appropriate initial trajectories for optimization, are analyzed deeply to enable previously unknown orbit options and potentially to reduce mission cost. A global search strategy based on comparing the orbital state of the unstable and stable manifolds, incorporated with low-thrust techniques, is performed to seek a suitable matching point for maneuver application. Then the sequential quadratic programming (SQP) is adopted to further optimize the velocity increment and obtain the single/multiple impulse optimal transfers. The associated constraint gradients are derived to achieve higher accuracy and rapidity of the algorithm. To highlight the effectivity of the transfer scheme, three-dimensional low-energy transfers between different types and spatial regions of performing single and multiple impulses are explored. The total Delta-V required varies between a few meters per second and tens of meters per second, and the related flight time is about several weeks, mainly depending on the energy of periodic orbits and the invariant manifold structure. The results obtained in this paper can provide a useful reference for the selection of escape and capture site along the manifolds, maneuver magnitude and transfer time.     

Mapping orbits around the asteroid 2001SN263

The present paper has the goal of mapping orbits, with respect to the perturbations, for a spacecraft traveling around the asteroid 2001SN₂₆₃. This asteroid is a triple system, which center of mass is in an elliptic orbit around the Sun. The perturbations considered in the present model are the ones due to the oblateness of the central body, the gravity field of the two satellite bodies (Beta and Gamma), the Sun, the Moon, the asteroids Vesta, Pallas and Ceres and all the planets of the Solar System. This mapping is important, because it shows the relative importance of each force for a given orbit for the spacecraft, helping to make a decision about which forces need to be included in the model for a given accuracy and nominal orbit. Another important application of this type of mapping is to find orbits that are less perturbed, since it is expected that those orbits have good potential to require a smaller number of station-keeping maneuvers. Simulations under different conditions are made to find those orbits. The main reason to study those trajectories is that, currently, there are several institutions in Brazil studying the possibility to make a mission to send a spacecraft to this asteroid (the so-called ASTER mission), because there are many important scientific studies that can be performed in that system. The results showed that Gamma is the main perturbing body, followed by Beta (10 times smaller) and the group Sun–Mars-oblateness of Alpha, with perturbations 1000 times weaker than the effects of Gamma. The other bodies have perturbations 10⁷ times smaller. The results also showed that circular and polar orbits are less perturbed, when compared to elliptical and equatorial orbits. Regarding the semi-major axis, an internal orbit is the best choice, followed by a larger external orbit. The inclination of the orbit plays an important role, and there are values for the inclination where the perturbations show minimum and maximum values, so it is important to make a good decision on those values.     

One dimensional blood flow in a planetocentric orbit

All life on earth is accustomed to the presence of gravity. When gravity is altered, biological processes can go awry. It is of great importance to ensure safety during a spaceflight. Long term exposure to microgravity can trigger detrimental physiological responses in the human body. Fluid redistribution coupled with fluid loss is one of the effects. In particular, in microgravity blood volume is shifted towards the thorax and head. Sympathetic nervous system-induced vasoconstriction is needed to maintain arterial pressure, while venoconstriction limits venous pooling of blood prevents further reductions in venous return of blood to the heart. In this paper, we modify an existing one dimensional blood flow model with the inclusion of the hydrostatic pressure gradient that further depends on the gravitational field modified by the oblateness and rotation of the Earth. We find that the velocity of the blood flow V_B is inversely proportional to the blood specific volume d, also proportional to the oblateness harmonic coefficient J₂, the angular velocity of the Earth ω_E, and finally proportional to an arbitrary constant c. For c = −0.39073 and ξ_H = −0.5 mmHg, all orbits result to less blood flow velocities than that calculated on the surface of the Earth. From all considered orbits, elliptical polar orbit of eccentricity e = 0.2 exhibit the largest flow velocity V_B = 1.031 m/s, followed by the orbits of inclination i = 45°and 0°. The Earth’s oblateness and its rotation contribute a 0.7% difference to the blood flow velocity.     

On two kinds of intermediate orbits for asteroid explorations

In the paper, two kinds of intermediate orbits for asteroid explorations are proposed. One is around the collinear libration points of the Sun-asteroid restricted three-body problem. The other is around the asteroid itself. The first kind of intermediate orbit is applicable to asteroids with known masses, while the second is suitable for asteroids with unknown or negligible masses. Analytical solutions of these two intermediate orbits in the simplified models are introduced first, and then numerical algorithms are used to refine them to obtain the true orbits in the real force model. At last, the problem of station-keeping is addressed. The linear optimal feedback control law is used, and numerical simulations are made to both kinds of intermediate orbits. The results show that both kinds of orbits are feasible. The cost is reasonable and mainly depends on the initial insertion error.     

The effect of geocenter motion on Jason-2 orbits and the mean sea level

We compute a series of Jason-2 GPS and SLR/DORIS-based orbits using ITRF2005 and the std0905 standards ( Lemoine et al., 2010). Our GPS and SLR/DORIS orbit data sets span a period of 2 years from cycle 3 (July 2008) to cycle 74 (July 2010). We extract the Jason-2 orbit frame translational parameters per cycle by the means of a Helmert transformation between a set of reference orbits and a set of test orbits. We compare the annual terms of these time-series to the annual terms of two different geocenter motion models where biases and trends have been removed. Subsequently, we include the annual terms of the modeled geocenter motion as a degree-1 loading displacement correction to the GPS and SLR/DORIS tracking network of the POD process. Although the annual geocenter motion correction would reflect a stationary signal in time, under ideal conditions, the whole geocenter motion is a non-stationary process that includes secular trends. Our results suggest that our GSFC Jason-2 GPS-based orbits are closely tied to the center of mass (CM) of the Earth consistent with our current force modeling, whereas GSFC’s SLR/DORIS-based orbits are tied to the origin of ITRF2005, which is the center of figure (CF) for sub-secular scales. We quantify the GPS and SLR/DORIS orbit centering and how this impacts the orbit radial error over the globe, which is assimilated into mean sea level (MSL) error, from the omission of the annual term of the geocenter correction. We find that for the SLR/DORIS std0905 orbits, currently used by the oceanographic community, only the negligence of the annual term of the geocenter motion correction results in a – 4.67 ± 3.40 mm error in the Z-component of the orbit frame which creates 1.06 ± 2.66 mm of systematic error in the MSL estimates, mainly due to the uneven distribution of the oceans between the North and South hemisphere.     

考虑太阳摄动的小行星附近轨道动力学

本文研究了艳后星(216 Kleopatra)和爱神星(433 Eros)附近的周期轨道,在考虑太阳引力摄动的情况下,发现了以往所遗漏的216 Kleopatra轨道族和环绕433 Eros的12族周期轨道,并且给出了它们的特性。研究结果表明,太阳引力对小行星平衡点位置的影响很小,但是对平衡点上航天器运动的影响较大。同族不稳定轨道中,大Jacobi常数轨道更容易在摄动后保持轨道原来特性,这很好地解释了小行星卫星在较远轨道上长期存在的可能性。     

Evolution of periodic orbits near the Lagrangian point L2

A study of the evolution of the periodic and the quasi-periodic orbits near the Lagrangian point L₂, which is located to the right of the smaller primary on the line joining the primaries and whose distance from the more massive primary is greater than the distance between the primaries, in the framework of restricted three-body problem for the Sun–Jupiter, Earth–Moon (relatively large mass ratio) and Saturn–Titan (relatively small mass ratio) systems is made. Two families of periodic orbits around the smaller primary are identified using the Poincaré surface of section method – family I (initially elliptical, gradually becomes egg-shaped with the increase in the Jacobi constant C and elongated towards the more massive primary) and family II (initially egg-shaped orbits elongated towards L₂ and gradually becomes elliptical with the increase in C). The family I in the Sun–Jupiter and Saturn–Titan systems contains two separatrix caused by third-order and fourth-order resonances, while the Earth–Moon system has only one separatrix which is caused by third-order resonances. Also in the Sun–Jupiter and the Saturn–Titan systems, family I merge with family II, around Jacobian constant 3.0393 and 3.0163, respectively, while in the Earth–Moon system, family II evolves separately from two different branches. The two branches merge at C = 3.184515. In the Earth–Moon system, the family II contains a separatrix due to third-order resonances which is absent in the other two systems.     

Estimation of the elastic Earth parameters (h2, l2) using SLR data

We present results for the global elastic parameters h₂ and l₂ derived from the analysis of Satellite Laser Ranging (SLR) data. SLR data for the two satellites LAGEOS 1 and LAGEOS 2 observed during 2.5 years from January 3, 2005 until July 1, 2007 with 18 globally distributed ground stations were analysed using different approaches. The analysis was done separately for the two satellites and approaches to estimate the two elastic parameters independently and together were performed. We do a sequential analysis and study the stability of the estimates as a function of length of the data set used. The adjusted final values for h₂ equal to 0.6151 ± 0.0008 and 0.6152 ± 0.0008, and those for l₂ equal to 0.0886 ± 0.0003 and 0.0881 ± 0.0003 for LAGEOS 1 and LAGEOS 2 tracking data are compared to other independently derived estimates. These parameters and their errors achieve stability at about the 24 and 27 month time interval for h₂ and l₂, respectively.     

地月平动点中继应用轨道维持

地月平动点中继应用轨道对于月球背面探测具有十分重要的应用价值,由于地月平动点的不稳定性,必须进行轨道维持。文章研究了真实力模型下月球平动点中继应用轨道的维持。首先,基于限制性三体问题下平动点轨道的运动特性,研究了平动点轨道维持的数学模型与维持策略,提出了平动点轨道维持的连续环绕控制方法,并给出了轨道维持的Halo和Lissajous两种控制方式;其次,充分考虑各天体和光压摄动下,采用数值手段研究了不同幅值的地月平动点周期中继应用轨道的维持间隔与速度增量等。研究结果表明:Lissajous控制方式适用于月球平动点中继应用轨道的维持,在给定测控精度条件下,维持间隔约7.4d,速度增量优于20m/s/a。该方法已经成功应用于我国"嫦娥2号"日地平动点任务和"嫦娥5T1"地月平动点任务并获得了良好的控制效果,还可直接应用于我国未来"嫦娥4号"等月球背面探测任务。     

Heliotropic dust rings for Earth climate engineering

This paper examines the concept of a Sun-pointing elliptical Earth ring comprised of dust grains to offset global warming. A new family of non-Keplerian periodic orbits, under the effects of solar radiation pressure and the Earth’s J₂ oblateness perturbation, is used to increase the lifetime of the passive cloud of particles and, thus, increase the efficiency of this geoengineering strategy. An analytical model is used to predict the orbit evolution of the dust ring due to solar-radiation pressure and the J₂ effect. The attenuation of the solar radiation can then be calculated from the ring model. In comparison to circular orbits, eccentric orbits yield a more stable environment for small grain sizes and therefore achieve higher efficiencies when the orbit decay of the material is considered. Moreover, the novel orbital dynamics experienced by high area-to-mass ratio objects, influenced by solar radiation pressure and the J₂ effect, ensure the ring will maintain a permanent heliotropic shape, with dust spending the largest portion of time on the Sun facing side of the orbit. It is envisaged that small dust grains can be released from a circular generator orbit with an initial impulse to enter an eccentric orbit with Sun-facing apogee. Finally, a lowest estimate of 1 × 10¹² kg of material is computed as the total mass required to offset the effects of global warming.