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.     

一种提升近地小行星防御中拦截效率的方法

利用核爆直接炸毁小行星或改变小行星的轨道以避免其与地球相撞,是近地小行星防御最主要的手段之一。文章基于美国爱荷华州立大学的超高速小行星拦截器(HAIV)概念,提出一种将原撞击引导器改为长杆撞击器的方案,采用自主研发的欧拉型冲击动力学仿真软件NTS模拟长杆撞击器对小行星连续开坑的过程,并在仿真中加入能量源以模拟核爆装置在不同深度爆炸对小行星产生的偏转与破坏效应。研究结果表明,采用长杆撞击器并合理控制撞击速度,能够引导核爆装置进入更深的地下爆炸,从而更加高效地耦合核爆能量,提升偏转小行星或直接摧毁小行星的能力。     

On analysis of close encounters of two cosmic bodies in near almost-circular orbits

This paper is devoted to the study of relative motion and close encounters of two cosmic bodies located in near almost-circular orbits. This problem is topical due to the asteroid hazard proceeding from the NEA group asteroids located in the near-Earth orbits. The (99942) Apophis asteroid, a representative of this group discovered in July 2004 by the Kitt Peak observatory (Arizona), is considered as an example.     

Experimental investigation of the modes of operation of uncontrolled attitude motion of the Progress spacecraft

Results of in-flight tests of three modes of uncontrolled attitude motion of the Progress spacecraft are described. These proposed modes of experiments related to microgravity are as follows: (1) triaxial gravitational orientation, (2) gravitational orientation of the rotating satellite, and (3) spin-up in the plane of the orbit around the axis of the maximum moment of inertia. The tests were carried out from May 24 to June 1, 2004 onboard the spacecraft Progress M1-11. The actual motion of this spacecraft with respect to its center of mass, in the above-mentioned modes, was determined by telemetric information about an electric current tapped off from solar batteries. The values of the current obtained during a time interval of several hours were processed jointly using the least squares method by integration of the equations of the spacecraft’s attitude motion. The processing resulted in estimation of the initial conditions of motion and of the parameters of mathematical models used. For the obtained motions the quasi-static component of microaccelerations was computed at a point onboard, where installation of experimental equipment is possible.     

Working and walking on small asteroids with circumferential ropes

The low gravity of a small asteroid would present a challenge for an astronaut attempting to work on its surface. Extravehicular activities (EVAs) of the sophistication of the Apollo Moon missions are not likely to be possible if astronauts attempt to walk freely on the asteroid, hover above its surface, or anchor locally into the regolith. Manipulating large rocks, drilling, and excavating at multiple locations is a high priority science objective, but would be difficult without a hold-down mechanism. If the asteroid has even a small rotation rate, maneuvering precisely over its surface could be cumbersome. A plausible means of conducting complex EVAs is to tie ropes entirely around the asteroid, under which the astronaut is pushed downward onto the asteroid surface by the tension in the rope. The downward force provides an artificial gravity that permits the astronaut to drill, excavate, hammer, and carefully document materials on the surface without the worry of being thrown from the asteroid. An astronaut could also use the ropes as handholds or guides to maneuver freely over the surface.     

Working and walking on small asteroids with circumferential ropes

The low gravity of a small asteroid would present a challenge for an astronaut attempting to work on its surface. Extravehicular activities (EVAs) of the sophistication of the Apollo Moon missions are not likely to be possible if astronauts attempt to walk freely on the asteroid, hover above its surface, or anchor locally into the regolith. Manipulating large rocks, drilling, and excavating at multiple locations is a high priority science objective, but would be difficult without a hold-down mechanism. If the asteroid has even a small rotation rate, maneuvering precisely over its surface could be cumbersome. A plausible means of conducting complex EVAs is to tie ropes entirely around the asteroid, under which the astronaut is pushed downward onto the asteroid surface by the tension in the rope. The downward force provides an artificial gravity that permits the astronaut to drill, excavate, hammer, and carefully document materials on the surface without the worry of being thrown from the asteroid. An astronaut could also use the ropes as handholds or guides to maneuver freely over the surface.     

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.     

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。     

Comparative investigation of methods for the numerical prediction of motion of asteroids that approach the Earth: Example of the 99942 Apophis asteroid

Using the 99942 Apophis asteroid (currently known as one of the most dangerous asteroids that is approaching the Earth) as an example, we estimate the error of predicting its motion with the use of several integrators over the time interval from 2012 to 2029. The minimum distance (and its error) between the Earth’s center and Apophis was estimated for the rendezvous moment on April 13, 2029. It is shown that this error for various integrators is comparable in the order of magnitude with the influence of certain components of the dynamic model of motion, such as, for example, the influence of harmonics of the Earth’s gravitational filed, solar-light pressure, the Jarkowski effect, etc.     

Landing on small bodies: From the Rosetta Lander to MASCOT and beyond

Recent planning for science and exploration missions has emphasized the high interest in the close investigation of small bodies in the Solar System. In particular in-situ observations of asteroids and comets play an important role in this field and will contribute substantially to our understanding of the formation and history of the Solar System.The first dedicated comet Lander is Philae, an element of ESA's Rosetta mission to comet 67/P Churyumov–Gerasimenko. Rosetta was launched in 2004. After more than 7 years of cruise (including three Earth and one Mars swing-by as well as two asteroid flybys) the spacecraft has gone into a deep space hibernation in June 2011. When approaching the target comet in early 2014, Rosetta will be re-activated. The cometary nucleus will be characterized remotely to prepare for Lander delivery, currently foreseen for November 2014.The Rosetta Lander was developed and manufactured, similar to a scientific instrument, by a consortium consisting of international partners. Project management is located at DLR in Cologne/Germany, with co-project managers at CNES (France) and ASI (Italy). The scientific lead is at the Max Planck Institute for Solar System Science (Lindau, Germany) and the Institut d'Astrophysique Spatiale (Paris).Mainly scientific institutes provided the subsystems, instruments and the complete, qualified lander system. Operations are performed in two dedicated centers, the Lander Control Center (LCC) at DLR-MUSC and the Science Operations and Navigation Center (SONC) at CNES. This concept was adopted to reduce overall cost of the project and is foreseen also to be applied for development and operations of future small bodies landers.A mission profiting from experience gained during Philae development and operations is MASCOT, a surface package for the Japanese Hayabusa 2 mission. MASCOT is a small (∼10 kg) mobile device, delivered to the surface of asteroid 1999JU3. There it will operate for about 16 h. During this time a camera, a magnetometer, a thermal monitor and an IR analytical instrument will provide ground truth and thus will even be able to support the selection of possible sampling sites for the main spacecraft.MASCOT is a flexible design that can be adapted to a wide range of missions and possible target bodies. Also the payload is flexible to some extent (with an overall mass in the 3 kg range). For example, the surface package is part of the optional strawman payload for MarcoPolo-R, a European asteroid sample return mission, proposed for ESA Cosmic Vision M-class.     

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。     

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.     

On coordinated approach to the problem of asteroid-comet impact hazard

A review of the current state of investigations into the complex global problem of asteroid-comet hazard (ACH) is presented. Main attention is drawn to a comprehensive analysis of prospects of developing the intra-Russia cooperation of studies on the ACH problem and initiating a working program of solving this problem on the national basis. Special space projects are briefly discussed, aimed to investigate and solve the ACH problem, in particular, the project of a mission of spacecraft with an onboard transponder to a hazardous asteroid (conventionally, asteroid Apophis is chosen as a target).     

System design of the Hayabusa 2—Asteroid sample return mission to 1999 JU3

The Japan Aerospace Exploration Agency is currently developing the second asteroid sample return mission, designated as Hayabusa 2. Following the successful return of Hayabusa from the asteroid “Itokawa”, Hayabusa 2 is designed as a round-trip mission to the asteroid “1999 JU3”. The 1999 JU3 is a C-type asteroid, which is believed to contain organic matter and hydrated minerals. Thus, it is expected that successful sample collection will provide additional knowledge on the origin and evolution of the planets and, in particular, the origin of water and organic matter. The current mission scenario will enable the spacecraft to reach 1999 JU3 in the middle of 2018 and perform an asteroid proximity operation for 1.5 years. Three touch downs for sampling and one 2-m-class crater generation by means of a high-speed impact operation are planned during the asteroid proximity operation. The samples are to be brought back to the Earth by a re-entry capsule. The present paper describes the system design of Hayabusa 2, some key technical challenges of the mission, and the development status.     

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。     

Structure of the Large-Scale Vortex Motions in the Ionosphere E Region

It is shown that the dissipative-centrifugal instability (DCI) revealed earlier for large-scale two-dimension vortex motions of the neutral gas in a fast-rotating atmosphere is also possible for similar motions of a weakly ionized gas in the magnetic field when one takes into account electrodynamics forces (the induction deceleration and gyroscopic force caused by the Hall current) in the absence of dissipation. In the case of a conductive atmosphere, strictly anticyclone motion at a DCI of the neutral atmosphere could be impossible or (depending on the Hall conductivity value) it could even change to a cyclone motion. Experimental results that can be considered as a confirmation of the theoretical conclusions are presented.     

Rosetta visits asteroid (21)Lutetia

The International Rosetta Mission, cornerstone of the European Space Agency Scientific Programme, was launched on 2nd March 2004 to its 10 years journey to comet Churyumov–Gerasimenko. Rosetta will reach the comet in summer 2014, orbit it for about 1.5 years down to distances of a few Kilometres and deliver the Lander Philae onto its surface. After its successful asteroid fly-by in September 2008, Rosetta came back to Earth, for the final gravity acceleration towards its longest heliocentric orbit, up to a distance of 5.3 AU. It is during this phase that Rosetta crossed for the second time the main asteroids belt and performed a close encounter with asteroid (21)Lutetia on the 10th of July 2010 at a distance of ca. 3160 km and a relative velocity of 15 km/s. The payload complement of the spacecraft was activated to perform highly valuable scientific observations. The approach phase to the celestial body required a careful and accurate optical navigation campaign that will prove to be useful also for the comet approach phase. The experience gained with first asteroid flyby in 2008 was fed back into the operations definition and preparation for this highly critical phase; this concerns in particular the operations of the navigation camera for the close-loop autonomous asteroid tracking and of the main scientific camera for high resolution imaging. It was shortly after the flyby that Rosetta became the solar-powered spacecraft to have flown furthest from the Sun (>2.72 AU). This paper presents the activities carried out and planned for the definition, preparation and implementation of the asteroid flyby mission operations, including the test campaign conducted to improve the performance of the spacecraft and payload compared to the previous flyby. The results of the flyby itself are presented, with the operations implemented, the achieved performance and the lessons learned.     

Optimal flights to near-Earth asteroid

The spacecraft flights to the Near-Earth asteroid in order to give an impact influence on the asteroid, correct its orbit and prevent the asteroid’s collision with the Earth are analyzed.In the first part, the impulse flights are analyzed in the Lambert approach. There are determined the optimal trajectories maximizing the asteroid deviation from the Earth.In the second part, the flights with the chemical and electric-jet engines are analyzed. The high thrust is used to launch the spacecraft from the geocentric orbit, and the low thrust is applied for the heliocentric motion. On the base of optimal impulse transfer, the optimal low thrust trajectories are determined using Pontryagin maximum principle.The numerical results are given for the flight to the asteroid Toutatis. Parameters of the spacecraft impact on the asteroid are determined. The asteroid deviation from the Earth caused by the spacecraft influence is presented.     

Impacts and evolution: future prospects

The discipline of astrobiology includes the dynamics of biological evolution. One of the major ways that the cosmos influences life is through the catastrophic environmental disruptions caused when comets and asteroids collide with a planet. We now recognize that such impacts have caused mass extinctions and played a major role in determining the evolution of life on Earth. The time-averaged impact flux as a function of projectile energy can be derived from lunar cratering statistics as well as the current population of near Earth asteroids (NEAs). Effects of impacts of various energies can be modeled, using data from historic impacts [such as the Cretaceous-Tertiary (KT) impactor 65 million years ago] and the observed 1994 bombardment of Jupiter by fragments of Comet Shoemaker-Levy 9. It is of particular interest to find from such models that the terrestrial environment is highly vulnerable to perturbation from impacts, so that even such a small event as the KT impact (by a projectile 10-15 km in diameter) can lead to a mass extinction. Similar considerations allow us to model the effects of still smaller (and much more likely) impacts, down to the size of the asteroid that exploded over Tunguska in 1908 (energy approximately 10 megatons). Combining the impact flux with estimates of environmental and ecological effects reveals that the greatest contemporary hazard is associated with impactors near 1 million megatons in energy (approximately 2 km in diameter for an asteroid). The current impact hazard is significant relative to other natural hazards, and arguments can be developed to illuminate a variety of public policy issues. The first priority in any plan for defense against impactors is to survey the population of Earth-crossing NEAs and project their orbits forward in time. This is the purpose of the Spaceguard Survey, which has already found more than half of the NEAs >1 km in diameter. If there is an NEA on a collision course with Earth, it can be discovered and the impact predicted with decades or more of warning. It is then possible to consider how to deflect or disrupt the NEA. Unlike other natural hazards, the impact risk can be largely eliminated, given sufficient advanced knowledge to take action against the threatening projectile.     

Deployment of a lander on the binary asteroid (175706) 1996 FG3, potential target of the european MarcoPolo-R sample return mission

The idea of deploying a lander on the secondary body of the binary primitive asteroid (175706) 1996 FG3 is investigated. 1996 FG3 is the backup target of the European sample return space mission MarcoPolo-R under assessment study at the European Space Agency in the framework of the M3 Medium-Class mission competition. The launch will take place in 2022–2024, depending on its selection at the end of 2013. A lander is indicated as an optional payload, depending on mass availability on the spacecraft. Obviously, the possible complexity of a lander deployment is also an important parameter to take into account. Here we demonstrate that, considering worst case scenarios and low requirements on the spacecraft GNC and deployment mechanism, the operations are easy to implement and safe for the main spacecraft. The concept of operations is to deploy a light lander from the L₂ Lagrange point of the binary system, on a ballistic trajectory that will impact the secondary asteroid. The fundamental principles of this strategy are briefly presented and a detailed model of 1996 FG3 is considered, to which the strategy is applied. We show that the deployment is successful in 99.94% of cases.