Kinematics of relative motion of two space bodies in close almost circular orbits

The paper is dedicated to a qualitative investigation of relative motion and close convergences of two space bodies located in close almost circular orbits. This problem is topical due to the asteroid hazard originating from the NEA group asteroids located in the orbits close to that of the Earth. P.E. El’yasberg [1] considered similar problems in the 1960s in relation to Earth’s artificial satellites in close almost circular orbits.     

同步双小行星系统共振轨道设计

研究了同步双小行星系统中共振轨道的设计方法及演化规律。首先，基于双椭球模型建立探测器运动方程，并给出共振轨道初值选取方法。然后，利用改进并行打靶法，提出一种双小行星系统平面共振轨道两步修正方法。同时结合稳定性理论及分岔理论，给出双小行星系统三维共振轨道生成和延拓方法；最后，以双小行星系统1999KW4为例，设计了共振比为1∶1,1∶2,1∶3,1∶4,2∶3的平面和空间共振轨道族，并分析了共振轨道的特性及轨道周期和轨道能量的变化规律。给出的双小行星系统中共振轨道的设计方法具有普适性，对未来双小行星系统探测任务中的轨道设计具有一定的参考意义与借鉴价值。     

航天器环绕小行星Ivar的运动分析

将小行星Ivar近似为三轴椭球体，给出了非球形引力势函数，建立了航天器环绕小行星Ivar的轨道动力学方程。利用Jacobi积分常数绘制了航天器在Ivar周围的零速度曲线，并分析了航天器的可能运动区域，给出了航天器不碰撞小行星Ivar的边界条件及不同偏心率下的近拱点半径。分析了小行星Ivar扁率和椭率对环绕轨道的影响，数学仿真结果表明：在一个轨道周期内，顺行轨道的开普勒能量、轨道角动量、偏心率和近拱点半径变化较大，而逆行轨道的相应参数变化较小。     

Scenarios which may lead to the rise of an asteroid-based technical civilisation

In a previous paper, the author described a hypothetical development path of technical civilisations which has the following stages: planet dwellers, asteroid dwellers, interstellar travellers, interstellar space dwellers. In this paper, several scenarios are described which may cause the rise of an asteroid-based technical civilisation. Before such a transition may take place, certain space technologies must be developed fully (now these exist only in very preliminary forms): closed-cycle biological life support systems, space manufacturing systems, electrical propulsion systems. After mastering these technologies, certain events may provide the necessary financial means and social impetus for the foundation of the first asteroid-based colonies. In the first scenario, a rich minority group becomes persecuted and they decide to leave the Earth. In the second scenario, a "cold war"-like situation exists and the leaders of the superpowers order the creation of asteroid-based colonies to show off their empires' technological (and financial) grandiosity. In the third scenario, the basic situation is similar to the second one, but in this case the asteroids are not just occupied by the colonists. With several decades of hard work, an asteroid can be turned into a kinetic energy weapon which can provide the same (or greater) threat as the nuclear arsenal of a present superpower. In the fourth scenario, some military asteroids are moved to Earth-centred orbits and utilised as "solar power satellites" (SPS). This would be a quite economical solution because a "military asteroid" already contains most of the important components of an SPS (large solar collector arrays, power distribution devices, orbit modifying rocket engine), one should add only a large microwave transmitter.     

State propagation in an uncertain asteroid gravity field

Various spacecraft have been and will be sent to asteroids to characterize them. Generally, an asteroid's gravity field is very irregular and not accurately known when compared to the gravity field of a major planet, Earth in particular. It has been well studied that the irregularity significantly affects the trajectory of an orbiting spacecraft, and causes it to impact or to escape from the asteroid. Complementary to that, this paper focuses on the influence of the limited knowledge of this gravity field on the evolution of the spacecraft's orbit. It develops a general method by which this influence can be quantified. This method comprises specific Monte Carlo simulations with a discrete set of low-altitude orbits, taking into account the uncertainties in the gravity-field parameters. For illustration purposes, it is applied to two different asteroids. Already after three revolutions, the gravity-field uncertainties propagate to significant position uncertainties; this specifically holds for prograde orbits, and around the smaller asteroid. Applying this robust and accurate method helps mission designers and planners to assess the risk posed by gravity uncertainties, and take appropriate measures such as choosing the most favorable orbital geometries and/or lowering the orbit more slowly.     

Dawn′s exploration of Vesta

On 16 July 2011, after completing nearly four years of interplanetary flight, Dawn entered orbit around (4) Vesta, the second most massive body in the main asteroid belt. Dawn used solar electric propulsion to spiral to six different orbits to accomplish its science campaign. Although the transfers to progressively lower orbits presented significant challenges, all were executed smoothly. During its nearly 14 months in orbit, Dawn spiraled down to 210 km above the surface and back up before initiating the gradual departure to travel to dwarf planet (1) Ceres for a 2015 rendezvous. Dawn′s exploration of Vesta has shown it to be geologically complex and fascinating, resembling terrestrial planets more than typical asteroids. Among the principal features is a 500-km-diameter impact basin within which is the second tallest mountain known in the solar system. This paper presents Dawn′s operations at Vesta and summarizes the principal findings.     

Analysis of the orbital motion of the asteroid Apophis’ satellite

We have analyzed the orbital disturbed spacecraft motion near an asteroid. The equations of the asteroidocentric spacecraft motion have been used with regard to three perturbations from celestial bodies, the asteroid’s nonsphericity, and solar radiation pressure. It has been shown that the orbital parameters of the main spacecraft and a small satellite with a radio beacon can be selected such that the orbits are rather stable for a fairly long period of time, i.e., a few weeks for the main spacecraft with an orbit initial radius of ~0.5 km and a few years before approaching Apophis with the Earth in 2029, for a small satellite at an orbit initial radius of ~1.5 km. The initial orientation of the spacecraft orbital plane perpendicular to the sunward direction is optimal from the point of view of the stability of the spacecraft flight near an asteroid.     

SETI implications of gravitational assist via chaotic trajectories of binary objects

Extrapolating from the technique of gravitational assist via chaotic trajectories of binary objects, this paper considers how such techniques might be used in other systems. We examine which types of systems are the best candidates for harvesting gravitational energy for payload ejection. We also consider what signatures might be present in either the asteroid orbits or radiation of the central body if extraterrestrial intelligences were to use such techniques about these candidate systems.The simulation studies show that current technology cannot approach the sensitivity needed to detect either of these signals. Instead, we provide these results as guidance to studies in coming decades on patterns that may indicate the use of an asteroid ejection system.     

One optimization problem for trajectories of spacecraft rendezvous mission to a group of asteroids

The optimization problem is considered for the trajectory of a spacecraft mission to a group of asteroids. The ratio of the final spacecraft mass to the flight time is maximized. The spacecraft is controlled by changing the value and direction of the jet engine thrust (small thrust). The motion of the Earth, asteroids, and the spacecraft proceeds in the central Newtonian gravitational field of the Sun. The Earth and asteroids are considered as point objects moving in preset elliptical orbits. The spacecraft departure from the Earth is considered in the context of the method of a point-like sphere of action, and the excess of hyperbolic velocity is limited. It is required sequentially to have a rendezvous with asteroids from four various groups, one from each group; it is necessary to be on the first three asteroids for no less than 90 days. The trajectory is finished by arrival at the last asteroid. Constraints on the time of departure from the Earth, flight duration, and final mass are taken into account in this problem.     

Equilibrium attitude and stability of a spacecraft on a stationary orbit around an asteroid

The stationary orbits around an asteroid, if exist, can be used for communication and navigation purposes just as around the Earth. The equilibrium attitude and stability of a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid are studied. The linearized equations of attitude motion are obtained under the small motion assumption. Then, the equilibrium attitude is determined in both cases of a general and a symmetrical spacecraft. Due to the higher-order inertia integrals of the spacecraft, the equilibrium attitude is slightly away from zero Euler angles. Then necessary conditions of stability of this conservative system are analyzed based on the linearized equations of motion. The effects of different parameters, including the harmonic coefficients C₂₀ and C₂₂ of the asteroid and higher-order inertia integrals of the spacecraft, on the stability are assessed and compared. Due to the significantly non-spherical shape and rapid rotation of the asteroid, the effects of the harmonic coefficients C₂₀ and C₂₂ are very significant, while effects of the third- and fourth-order inertia integrals of the spacecraft can be neglected. Considering a spacecraft on a stationary orbit around an example asteroid, we show that the classical stability domain predicted by the Beletskii–DeBra–Delp method on a circular orbit in a central gravity field is modified due to the non-spherical mass distribution of the asteroid. Our results are confirmed by a numerical simulation.     

Asteroidal resources for space manufacturing

Earth-approaching asteroids (Apollos and Amors) may be competitive candidates as raw materials for space manufacturing. The total energy per unit mass required to transfer material from some of these bodies to high Earth orbit is comparable to that for lunar material. Recent optical studies suggest ordinary and carbonaceous chondrite compositions for these asteroids, with some containing large quantities of metallic iron and nickel, and others, carbon, hydrogen and nitrogen. Discoveries of several new candidate asteroids over the next few years will allow for a better selection of materials and mission possibilities. Material from one of these asteroids, either in raw or manufactured form, could be returned to the vicinity of the Earth by a solar-powered mass-driver reaction engine. With a requirement of ～60 shuttle flights, and with minimal development costs, an automated mission to a 200-m dia. (10⁷ ton) metal-rich asteroid could be carried out by a mass-driver tug assembled in low Earth orbit using shuttle tankage as reaction mass. Such a tug could, within a few years, move the asteroid into high Earth orbit for the manufacturing of ～ 20 satellite power stations using a portion of the asteroid itself as reaction mass. In the next few years over 100 asteroids in this size range could be discovered, orbits determined and composition types classified using existing earthbased and spaceborne search techniques.     

On plane oscillations of a pendulum with variable length suspended on the surface of a planet’s satellite

The problem of planar oscillations of a pendulum with variable length suspended on the Moon’s surface is considered. It is assumed that the Earth and Moon (or, in the general case, a planet and its satellite, or an asteroid and a spacecraft) revolve around the common center of mass in unperturbed elliptical Keplerian orbits. We discuss how the change in length of a pendulum can be used to compensate its oscillations. We wrote equations of motion, indicated a rule for the change in length of a pendulum, at which it has equilibrium positions relative to the coordinate system rotating together with the Moon and Earth. We study the necessary conditions for the stability of these motions. Chaotic dynamics of the pendulum is studied numerically and analytically.     

小行星探测最优两脉冲交会轨道设计与分析

小行星探测已经成为新世纪深空探测的一个新热点和未来世界航天发展的一个新方向。转移轨道的设计和探测目标可接近性的分析是小行星探测的关键技术之一。现利用了任意两个非共面非共轴椭圆轨道之间的最优两脉冲转移方法，对我国提出的探测Ivar小行星的交会转移轨道进行了设计与分析，给出了全局最优两脉冲交会轨道的设计参数，并利用此方法对近地小行星的可接近性进行了分析和排序，给出了可接近性较好的40颗近地小行星的转移轨道设计参数。这些研究结果对于近地小行星探测任务的目标选择和发射机会的预测都有重要的参考价值。     

不规则形状小行星引力环境建模及 球谐系数求取方法

在实施小行星探测任务之前,需要对不规则形状小行星的引力场有一个清楚的认识,以便于进行环绕或着陆小行星的轨道设计。文章提出了一种小行星引力场建模及球谐系数的求取方法。首先,由多面体模型方法重构出小行星外部的引力环境并以此作为虚拟观测量。在此基础上,根据球谐系数与引力势能间的关系,通过求解超定线性方程组得到小行星引力场的各阶次球谐系数。与传统的将小行星近似成三轴椭球体进而计算球谐系数的方法相比,该方法可大幅度提高引力场建模的精度。通过与由NEAR探测器轨道数据解算的Eros433小行星的球谐系数比较表明,其最大误差不超过6%。计算表明,该方法可以为小行星探测任务发射前的轨道设计提供更为精确的数据。     

Space missions for SETI.

Radio Telescopes for SETI searches are less demanding than general purpose astronomical radio telescopes. This provides an opportunity to exploit economical approaches in designing SETI systems. Radio Telescopes in low Earth orbit offer no discernible advantages to SETI; indeed, they probably would perform more poorly than a telescope in any other location. Telescopes in geosynchronous orbits would be sufficiently far from Earth to mitigate greatly the deleterious effect of human radio transmissions. Telescopes on the far side of the moon would be superb both from a radio interference standpoint, and from a civil engineering standpoint. Single-reflector telescopes as large as 50 kilometers in diameter could be constructed with conventional materials. However, their costs appear prohibitive. The asteroid belt and the outer solar system are unpromising places to place a large radio telescope. Perhaps the ultimate radio telescope would utilize the sun as a gravitational lens, focusing radiation on free-flying 10-meter class or possibly larger radio telescopes located at distances of the order of 1000 A.U. from the sun. Such a combination has an energy collecting area at 10 centimeters wavelength equivalent to that of a radio telescope about 11 kilometers in diameter, or of the order of 3000 Arecibo radio telescopes. Such a system could detect transmitters with EIRP of the order of a gigawatt at a distance of the order of the distance to the galactic center.     

序列凸优化的小天体附着轨迹优化

针对小天体附着多约束轨迹优化问题,提出一种基于序列凸优化的轨迹优化方法。首先采用内球谐引力场模型对目标小天体附近的不规则引力场进行精确建模,内球谐引力场模型是对经典球谐系数法的改进,形式简单,计算量小,并且克服了经典球谐系数模型在形状不规则的小天体附近不收敛的问题。对于小天体附着多约束轨迹优化问题,通过约束松弛、线性化、离散化过程,转化为一个可以迭代求解的二阶锥规划问题（SOCP）,进而采用内点法进行解算。数学仿真结果显示,优化结果符合各项约束条件,以零速度到达了目标着陆点,且符合燃耗最优的优化目标。利用序列凸优化算法进行小天体附着燃耗最优轨迹设计,推导简便,计算速度快,精度高,具有应用价值。     

载人小行星探测的任务特点与实施途径探讨

介绍了载人小行星探测的发展现状,对目前美国基于"猎户座"飞船的载人小行星探测的概要方案进行了描述,包括探测器系统组成、运载火箭和飞行方案等内容。从速度增量、目标星引力等方面,分析了载人小行星探测的任务特点,并与载人火星探测、载人月球探测以及无人小行星探测的任务特点进行了比较。给出了载人小行星探测的实施途径建议,包括目标星选择、载人飞船系统设计等。讨论了其所涉及的推进、星际飞行安全保障、小行星表面行走等关键技术。研究结果可为我国开展载人深空探测提供参考。     

考虑地球扁率效应的两圆轨道之间的双共切最优转移

两圆轨道之间的双共切转移轨道是其近地点和远地点分别在这两个圆轨道上的椭圆轨道。本文用两次冲量法给出了沿双共切椭圆轨道实现从一圆轨道向另一圆轨道转移的最优方案，并考虑到地球扁率造成的轨道摄动。文中的所谓圆轨道指的是变轨时刻的密切轨道为圆形的轨道，是对近圆轨道的近似替找。     

Generalized restricted circular three-body problem as a model for dynamics of binary asteroids

Exploration of the Solar System has recently revealed the existence of a large number of asteroids with satellites, which has stimulated interest in studying the dynamics of such systems. This paper is dedicated to the analysis of the relative motion of a binary asteroid. The conditions of existence of such a system (i.e., when its components do not run away) are derived in the Introduction. Then it is assumed that the satellite has no significant effect on the motion of the main asteroid, the latter being modeled as a dumbbell-like precessing solid body. The equations of motion of this system are a two-parameter generalization of the corresponding equations of the restricted circular three-body problem. It is demonstrated that in the system under consideration there exist steady-state motions in which the small asteroid is equidistant from attracting centers at the ends of the dumbbell (an analog to triangle libration points). The conditions of existence of such motions are derived, and the positions with respect to the dumbbell are analyzed in detail. Examination of the stability of the triangle libration points is reduced to investigation of a characteristic equation of the sixth degree. The stability conditions are derived in the case when the main asteroid executes near-planar motion.     

The Aster project: Flight to a near-Earth asteroid

The information on the project being developed in Brazil for a flight to binary or triple near-Earth asteroid is presented. The project plans to launch a spacecraft into an orbit around the asteroid and to study the asteroid and its satellite within six months. Main attention is concentrated on the analysis of trajectories of flight to asteroids with both impulsive and low thrust in the period 2013-2020. For comparison, the characteristics of flights to the (45) Eugenia triple asteroid of the Main Belt are also given.