Optimal station-keeping near Earth–Moon collinear libration points using continuous and impulsive maneuvers

In this study the gravitational perturbations of the Sun and other planets are modeled on the dynamics near the Earth–Moon Lagrange points and optimal continuous and discrete station-keeping maneuvers are found to maintain spacecraft about these points. The most critical perturbation effect near the L₁ and L₂ Lagrange points of the Earth–Moon is the ellipticity of the Moon’s orbit and the Sun’s gravity, respectively. These perturbations deviate the spacecraft from its nominal orbit and have been modeled through a restricted five-body problem (R5BP) formulation compatible with circular restricted three-body problem (CR3BP). The continuous control or impulsive maneuvers can compensate the deviation and keep the spacecraft on the closed orbit about the Lagrange point. The continuous control has been computed using linear quadratic regulator (LQR) and is compared with nonlinear programming (NP). The multiple shooting (MS) has been used for the computation of impulsive maneuvers to keep the trajectory closed and subsequently an optimized MS (OMS) method and multiple impulses optimization (MIO) method have been introduced, which minimize the summation of multiple impulses. In these two methods the spacecraft is allowed to deviate from the nominal orbit; however, the spacecraft trajectory should close itself. In this manner, some closed or nearly closed trajectories around the Earth–Moon Lagrange points are found that need almost zero station-keeping maneuver.     

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.     

Model of the all-sky He II 30.4 nm solar flux

Because of the orbit characteristics of the vast majority of spacecraft, the solar flux has been generally measured at Earth or in the plane of the ecliptic. So far, most published studies did not consider the fact that the extreme ultraviolet (EUV) solar flux is largely anisotropic. Indeed, in the EUV, the distribution of very contrasted bright and dark features at the surface of the Sun produces both the obvious rotational (longitudinal) modulation of the flux, but also a strong latitudinal anisotropy. Although largely ignored up to now, the latitudinal anisotropy affects the physical conditions in the corona and heliosphere. We describe an empirical model of the all-sky He II 30.4 nm flux based on EIT/SOHO data. The 30.4 nm flux was found to be strongly anisotropic. The anisotropy I_pol/I_eq between the fluxes computed for viewpoints located above the solar poles and within the solar equatorial plane ranges from 0.9 at solar minimum to 0.6 at solar maximum. A 20% asymmetry was also discovered between the north and south polar fluxes.     

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.     

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.     

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.     

Effects of in-track maneuver on Sun illumination conditions of near-circular low Earth orbits

By developing approximate analytical models considering the J₂ perturbation, the effects of an in-track maneuver on the orbital Sun illumination conditions of near-circular low Earth orbits are analyzed. First, two approximate models for the variations in orbital sunshine angles are developed, one for variations at a given time and the other for variations at a given argument of latitude. Next, two approximate models for variations in orbital arc in Earth shadow are developed, one considers the small eccentricity and the other uses the zero eccentricity. Finally, the developed approximate models are applied to analyzing the Sun illumination conditions of a typical in-track maneuver mission on a near-circular low Earth orbit. From the results obtained, three major conclusions can be drawn. First, the variations in orbital sunshine angles at a given time may reach tens of degrees when the drifting time reaches hundreds of orbital periods, and the approximate model for that situation cannot effectively approach the numerical results. Second, the variations in orbital sunshine angles for any given argument of latitude are only a couple of degrees even when the drifting time reaches 500 orbital periods, and the approximation model developed can effectively approach the numerical results. Third, for variations in orbital arc in Earth shadow, the approximate model considering the small eccentricity has simple expressions and can effectively approach the numerical results; in contrast, the approximate model using the zero eccentricity has relatively worse precision.     

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.     

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.     

Resonant quasi-periodic near-rectilinear Halo orbits in the Elliptic-Circular Earth-Moon-Sun Problem

Motivated by the near-future re-exploration of the cislunar space, this paper investigates dynamical substitutes of the Earth-Moon’s resonant Near-Rectilinear Halo Orbits (NRHOs) under the Elliptic-Circular Restricted Four-Body Problem formulation of the Earth-Moon-Sun system. This model considers that the Earth and Moon move in elliptical orbits about each other and that a third body, the Sun, moves in a circular orbit about the Earth-Moon barycenter. By making use of this higher-fidelity dynamical model, we are able to incorporate the Sun’s influence and the Moon’s eccentricity, two of the most significant perturbations of the cislunar environment. As a result of these perturbations, resonant periodic NRHOs of the Earth-Moon Circular Restricted Three-Body Problem (CR3BP) are hereby replaced by two-dimensional quasi-periodic tori that better represent the dynamical evolution of satellites near the vicinity of the Moon. We present the steps and algorithms needed to compute these dynamical structures in the Elliptic-Circular model and subsequently assess their utility for spacecraft missions. We focus on the planned orbit for the NASA-led Lunar Gateway mission, a 9:2 synodic resonant L₂ southern NRHO, as well as on the 4:1 synodic and 4:1 sidereal resonances, due to the proximity to the nominal orbit and their advantageous dynamical properties. We verify that the dynamical equivalents of these orbits preserve key dynamical attributes such as eclipse avoidance and near-linear stability. Furthermore, we find that the higher dimensionality of quasi-periodic solutions offers interesting alternatives to mission designers in terms of phasing maneuvers and low-altitude scientific observations.     

Sun earth connection coronal and heliospheric investigation (SECCHI)

The Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) on the NASA Solar Terrestrial Relations Observatory (STEREO) mission is a suite of remote sensing instruments consisting of an extreme ultraviolet imager, two white light coronagraphs, and a heliospheric imager. Two spacecraft with identical instrumentation will obtain simultaneous observations from viewpoints of increasing separation in the ecliptic plane. In support of the STEREO mission objectives, SECCHI will observe coronal mass ejections from their birth at the Sun, through the outer corona, to their impact at Earth. The SECCHI program includes a coordinated effort to develope magneto-hydrodynamic models and visualization tools to interpret the images that will be obtained from the two spacecraft viewpoints. The resulting three-dimensional analysis of CMEs will help to resolve some of the fundamental outstanding questions in solar physics.     

北斗卫星导航系统X射线脉冲星可见性分析

X射线脉冲星可以为深空乃至近地航天器提供全源、自主、高精度的导航服务.对近地航天器而言，由于中心天体的遮挡，脉冲星信号并非全程可见，所以X射线脉冲星的可见性成为近地导航应用中挑选脉冲星的关键因素，也是脉冲星导航的首要工作.本文对挑选出的30颗X射线脉冲星在北斗系统7个轨道面上的可见性进行了仿真分析，得出了一天内可见脉冲星数量的变化情况以及一年内太阳、地球和月球遮挡的仿真结果,总结了影响脉冲星可见性的天体的遮挡规律，为利用脉冲星对北斗卫星进行自主定轨和守时提供了数据支持.     

Multiobjective genetic optimization of Earth–Moon trajectories in the restricted four-body problem

In this paper, optimal trajectories of a spacecraft traveling from Earth to Moon using impulsive maneuvers (ΔV maneuvers) are investigated. The total flight time and the summation of impulsive maneuvers ΔV are the objective functions to be minimized. The main celestial bodies influencing the motion of the spacecraft in this journey are Sun, Earth and Moon. Therefore, a three-dimensional restricted four-body problem (R4BP) model is utilized to represent the motion of the spacecraft in the gravitational field of these celestial bodies. The total ΔV of the maneuvers is minimized by eliminating the ΔV required for capturing the spacecraft by Moon. In this regard, only a mid-course impulsive maneuver is utilized for Moon ballistic capture. To achieve such trajectories, the optimization problem is parameterized with respect to the orbital elements of the ballistic capture orbits around Moon, the arrival date and a mid-course maneuver time. The equations of motion are solved backward in time with three impulsive maneuvers up to a specified low Earth parking orbit. The results show high potential and capability of this type of parameterization in finding several Pareto-optimal trajectories. Using the non-dominated sorting genetic algorithm with crowding distance sorting (NSGA-II) for the resulting multiobjective optimization problem, several trajectories are discovered. The resulting trajectories of the presented scheme permit alternative trade-off studies by designers incorporating higher level information and mission priorities.     

Yarkovsky-Schach effect on space debris motion

The Yarkovsky-Schach effect is a small perturbation affecting Earth satellites and space debris illuminated by the Sun. It was first applied to the orbit of LAGEOS satellites as an explanation of the residuals in orbital elements. In this work, we carry out several numerical integration tests taking into consideration various orbit and rotation parameters, in order to analyse this effect in a broader context. The semi-major axis variations remain small and depend on the spin axis attitude with respect to the Sun. We show that the force amplitude is maximised for orbits inclined with $i\approx$?20–30°. We also observe the influence on other orbital elements, notably on the orbit inclination. However, these effects are clearly observed only on long timescales; in our simulations, we propagated the orbits for 200?y. The Yarkovsky-Schach effect is thus confirmed to have a minuscule magnitude. It should be taken into account in studies requiring high-precision orbit determination, or on expanded timescales.     

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.     

On the effects of each term of the geopotential perturbation along the time I: Quasi-circular orbits

This paper provides a useful new method to determine minimum and maximum range of values for the degree and order of the geopotential coefficients required for simulations of orbits of satellites around the Earth. The method consists in a time integration of the perturbing acceleration coming from each harmonic of the geopotential during a time interval T. More precisely, this integral represents the total velocity contribution of a specific harmonic during the period T  . Therefore, for a pre-fixed minimum contribution, for instance 1×10^-8$1\times {10}^{-8}$ m/s during the period of time T, any harmonic whose contribution is below this value can, safely, be neglected. This fact includes some constraints in the degree and order of the terms which are present in the geopotential formula, saving computational efforts compared to the integration of the full model. The advantage of this method is the consideration of other perturbations in the dynamics (we consider the perturbations of the Sun, the Moon, and the direct solar radiation pressure with eclipses), since these forces affect the value of the perturbation of the geopotential, because these perturbations depend on the trajectory of the spacecraft, that is dependent on the dynamical model used. In this paper, we work with quasi-circular orbits and we present several simulations showing the bounds for the maximum degree and order (M) that should be used in the geopotential for different situations, e. g., for a satellite near 500 km of altitude (like the GRACE satellites at the beginning of their mission) we found 35?M?198$35?M?198$ for T=1$T=1$ day. We analyzed the individual contribution of the second order harmonic (_J2$J_2$) and we use its behavior as a parameter to determine the lower limit of the number of terms of the geopotential model. In order to test the accuracy of our truncated model, we calculate the mean squared error between this truncated model and the “full” model, using the CBERS (China-Brazil Earth Resources Satellite) satellite in this test.     

从日地系统L2出发借力月球飞越近地小行星

对于停留在日地系统L2的“嫦娥2号”探测器,其后续飞行方案有多个选项,例如主动撞月或重返月球轨道、返回地球轨道或再入大气、飞往地月系统L1/L2或日地系统L1、进入深空飞越近地小行星(最终,“嫦娥2号”于2012年12月13日成功地实现了对Toutatis小行星的近距离飞越)。探讨上述的飞行方案需要对飞行轨道进行初步设计,总的速度脉冲限制在100 m/s以内并且需要考虑探测器同时受到太阳、地球、月球的引力作用。本研究设计了探测器从日地系统L2出发借力月球实现Toutatis小行星飞越的飞行方案,与直接飞越方案相比,借力月球可以进一步节省探测器的燃料消耗,其等效速度脉冲设计值为58.47 m/s。     

Halo轨道探测器的姿态描述与建模

日地系Halo轨道位于平动点附近,且不存在以地球为中心的轨道平面,这使得Halo轨道探测器的姿态描述和建模问题完全不同于近地卫星.针对三轴稳定方式的探测器,给出了2种姿态描述方案:对日定向方案(方案I,基于日、地敏感器)和旋转坐标系下定向方案(方案II,基于星敏感器).依据两者所定义的不同轨道坐标系,分别建立了姿态动力学模型.分析2类描述方案的相对误差,发现方案I的上限不超过5%,而方案II的上限不超过0.005%.并就方案I设计了姿态镇定控制器,以验证所提出的姿态描述方案的合理性.结果表明:具体工程应用中应以方案II为正常工作模式,并定期修正星历表的误差;方案I作为备份,保证探测器的正常运行.      

Solar orbiter, a high-resolution mission to the sun and inner heliosphere

The scientific rationale of the Solar Orbiter is to provide, at high spatial (35 km pixel size) and temporal resolution, observations of the solar atmosphere and unexplored inner heliosphere. Novel observations will be made in the almost heliosynchronous segments of the orbits at heliocentric distances near 45 R and out of the ecliptic plane at the highest heliographic latitudes of 30° – 38°. The Solar Orbiter will achieve its wide-ranging aims with a suite of sophisticated instruments through an innovative design of the orbit. The first near-Sun interplanetary measurements together with concurrent remote observations of the Sun will permit us to determine and understand, through correlative studies, the characteristics of the solar wind and energetic particles in close linkage with the plasma and radiation conditions in their source regions on the Sun. Over extended periods the Solar Orbiter will deliver the first images of the polar regions and the side of the Sun invisible from the Earth.