Optimization of transfer trajectories to the Apophis asteroid for spacecraft with high and low thrust

The problem of optimization of a spacecraft transfer to the Apophis asteroid is investigated. The scheme of transfer under analysis includes a geocentric stage of boosting the spacecraft with high thrust, a heliocentric stage of control by a low thrust engine, and a stage of deceleration with injection to an orbit of the asteroid’s satellite. In doing this, the problem of optimal control is solved for cases of ideal and piecewise-constant low thrust, and the optimal magnitude and direction of spacecraft’s hyperbolic velocity “at infinity” during departure from the Earth are determined. The spacecraft trajectories are found based on a specially developed comprehensive method of optimization. This method combines the method of dynamic programming at the first stage of analysis and the Pontryagin maximum principle at the concluding stage, together with the parameter continuation method. The estimates are obtained for the spacecraft’s final mass and for the payload mass that can be delivered to the asteroid using the Soyuz-Fregat carrier launcher.     

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).     

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.     

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.     

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.     

Investigation of the motion of (99942) apophis asteroid using the SKIF cyberia multiprocessor computing system

We present the results of investigation of the dynamics of (99942) Apophis asteroid, which will undergo a very close encounter with the Earth on April 13, 2029. The region of possible motions of the asteroid is considered on the time interval (2004, 2040). In addition, it is shown that an increase of the observational interval (2004, 2006) until 2008 allowed us to reduce significantly the area of possible motions. All investigations were performed by numerical methods with the help of algorithms and software developed by us in the environment of parallel programming using the SKIF Cyberia multiprocessor computer of the Tomsk State University.     

A mission template for exploration and damage mitigation of potential hazard of Near Earth Asteroids

The Apophis Exploratory and Mitigation Platform (AEMP) concept was developed as a prototype mission to explore and potentially deflect the Near Earth Asteroid (NEA) 99942 Apophis. Deflection of the asteroid from the potential 2036 impact will be achieved using a gravity tractor technique, while a permanent deflection, eliminating future threats, will be imparted using a novel albedo manipulation technique. This mission will serve as an archetypal template for future missions to small NEAs and could be adapted to mitigate the threat of collision with other potential Earth-crossing objects.     

Fast solar sail rendezvous mission to near Earth asteroids

The concept of fast solar sail rendezvous missions to near Earth asteroids is presented by considering the hyperbolic launch excess velocity as a design parameter. After introducing an initial constraint on the hyperbolic excess velocity, a time optimal control framework is derived and solved by using an indirect method. The coplanar circular orbit rendezvous scenario is investigated first to evaluate the variational trend of the transfer time with respect to different hyperbolic excess velocities and solar sail characteristic accelerations. The influence of the asteroid orbital inclination and eccentricity on the transfer time is studied in a parametric way. The optimal direction and magnitude of the hyperbolic excess velocity are identified via numerical simulations. The found results for coplanar circular scenarios are compared in terms of fuel consumption to the corresponding bi-impulsive transfer of the same flight time, but without using a solar sail. The fuel consumption tradeoff between the required hyperbolic excess velocity and the achievable flight time is discussed. The required total launch mass for a particular solar sail is derived in analytical form. A practical mission application is proposed to rendezvous with the asteroid 99942 Apophis by using a solar sail in combination with the provided hyperbolic excess velocity.     

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。     

Planetary Defense From Space: Part 2 (Simple) Asteroid Deflection Law

A system of two space bases housing missiles for an efficient Planetary Defense of the Earth from asteroids and comets was firstly proposed by this author in 2002. It was then shown that the five Lagrangian points of the Earth–Moon system lead naturally to only two unmistakable locations of these two space bases within the sphere of influence of the Earth. These locations are the two Lagrangian points L1 (in between the Earth and the Moon) and L3 (in the direction opposite to the Moon from the Earth). In fact, placing missiles based at L1 and L3 would enable the missiles to deflect the trajectory of incoming asteroids by hitting them orthogonally to their impact trajectory toward the Earth, thus maximizing the deflection at best. It was also shown that confocal conics are the only class of missile trajectories fulfilling this “best orthogonal deflection” requirement.The mathematical theory developed by the author in the years 2002–2004 was just the beginning of a more expanded research program about the Planetary Defense. In fact, while those papers developed the formal Keplerian theory of the Optimal Planetary Defense achievable from the Earth–Moon Lagrangian points L1 and L3, this paper is devoted to the proof of a simple “(small) asteroid deflection law” relating directly the following variables to each other:

(1) the speed of the arriving asteroid with respect to the Earth (known from the astrometric observations);

(2) the asteroid's size and density (also supposed to be known from astronomical observations of various types);

(3) the “security radius” of the Earth, that is, the minimal sphere around the Earth outside which we must force the asteroid to fly if we want to be safe on Earth. Typically, we assume the security radius to equal about 10,000 km from the Earth center, but this number might be changed by more refined analyses, especially in the case of “rubble pile” asteroids;

(4) the distance from the Earth of the two Lagrangian points L1 and L3 where the defense missiles are to be housed;

(5) the deflecting missile's data, namely its mass and especially its “extra-boost”, that is, the extra-energy by which the missile must hit the asteroid to achieve the requested minimal deflection outside the security radius around the Earth.

This discovery of the simple “asteroid deflection law” presented in this paper was possible because:

(1) In the vicinity of the Earth, the hyperbola of the arriving asteroid is nearly the same as its own asymptote, namely, the asteroid's hyperbola is very much like a straight line. We call this approximation the line/circle approximation. Although “rough” compared to the ordinary Keplerian theory, this approximation simplifies the mathematical problem to such an extent that two simple, final equations can be derived.

(2) The confocal missile trajectory, orthogonal to this straight line, ceases then to be an ellipse to become just a circle centered at the Earth. This fact also simplifies things greatly. Our results are thus to be regarded as a good engineering approximation, valid for a preliminary astronautical design of the missiles and bases at L1 and L3.

Still, many more sophisticated refinements would be needed for a complete Planetary Defense System:

(1) taking into account many perturbation forces of all kinds acting on both the asteroids and missiles shot from L1 and L3;

(2) adding more (non-optimal) trajectories of missiles shot from either the Lagrangian points L4 and L5 of the Earth–Moon system or from the surface of the Moon itself;

(3) encompassing the full range of missiles currently available to the USA (and possibly other countries) so as to really see “which missiles could divert which asteroids”, even just within the very simplified scheme proposed in this paper.

In summary: outlined for the first time in February 2002, our Confocal Planetary Defense concept is a simplified Keplerian Theory that already proved simple enough to catch the attention of scholars, popular writers, and representatives of the US Military. These developments would hopefully mark the beginning of a general mathematical vision for building an efficient Planetary Defense System in space and in the vicinity of the Earth, although not on the surface of the Earth itself!We must make a real progress beyond academic papers, Hollywood movies and secret military plans, before asteroids like 99942 Apophis get close enough to destroy us in 2029 or a little later.     

An autonomous optical guidance and navigation around asteroids

An impending demand for exploring the small bodies such as the comets and the asteroids envisioned the Japanese MUSES-C mission to the near Earth asteroid Nereus. An autonomous optical guidance and navigation strategy around the asteroid is discussed in this paper. Four major new schemes are dealt with here: They are (1) Aligned intercept guidance, (2) Strategic building of the flight phases, (3) Image processing of line-of-sight shift information instead of characteristic point tracking, and (4) Stability and accuracy analysis associated with the guidance and navigation strategies developed here. Some comprehensive numerical illustrations are also given to support them.     

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。     

Assessment on the feasibility of future shepherding of asteroid resources

Most plausible futures for space exploration and exploitation require a large mass in Earth orbit. Delivering this mass requires overcoming the Earth's natural gravity well, which imposes a distinct obstacle to any future space venture. An alternative solution is to search for more accessible resources elsewhere. In particular, this paper examines the possibility of future utilisation of near Earth asteroid resources. The accessibility of asteroid material can be estimated by analysing the volume of Keplerian orbital element space from which Earth can be reached under a certain energy threshold and then by mapping this analysis onto an existing statistical near Earth objects (NEO) model. Earth is reached through orbital transfers defined by a series of impulsive manoeuvres and computed using the patched-conic approximation. The NEO model allows an estimation of the probability of finding an object that could be transferred with a given Δv budget. For the first time, a resource map provides a realistic assessment of the mass of material resources in near Earth space as a function of energy investment. The results show that there is a considerable mass of resources that can be accessed and exploited at relatively low levels of energy. More importantly, asteroid resources can be accessed with an entire spectrum of levels of energy, unlike other more massive bodies such as the Earth or Moon, which require a minimum energy threshold implicit in their gravity well. With this resource map, the total change of velocity required to capture an asteroid, or transfer its resources to Earth, can be estimated as a function of object size. Thus, realistic examples of asteroid resource utilisation can be provided.     

The use of radio images for analyzing the results of narrow-band radio location of asteroids

We investigate a method of getting radio images of asteroids by single-position radar with narrowband sounding signal. A model is presented for calculation of power spectra of echo signals of a rotating rough ellipsoid. This model is a useful tool for analyzing the results of radio-location experiments. The accuracy of determination of the shape of an asteroid and its roughness is estimated based on the analysis of two-dimensional radio images.     

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.     

Optimal thrust program for deviating an Earth-pointing asteroid

The purpose of this study is to investigate the optimal thrust program for deviating an asteroid when it is flying directly toward or crossing the Earth. Under some assumptions, the problem can be considered as a two-body problem of Earth-asteroid system. The initial relative speed and distance are specified to be 10 km/s and 150 times of Earth's radius, respectively. We have about 1 day (or, 93681 seconds exactly) to take action. If a single impulse is applied to the asteroid at the specified initial point, the required impulse to obtain a miss distance of 2 times of Earth's radius is 169.5 m/s per kg mass. For an asteroid of 10 m in diameter, the total impulse required is 3.02 × 10⁸ m/s. It needs a typical large launching rocket to provide the total impulse. When the asteroid is larger or the initial distance is shorter, the number of launching rockets required increases rapidly. For further analysis with physical and engineering constraints imposed, we shall have to use the variational formulation method.     

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.     

Distribution of Plasma Pressure in the Geomagnetic Tail in the Transition Region from Dipole to Quasidipole and Stretched Magnetic Field Lines: Events on October 13, 1995 and March 13, 1996

The results of an analysis of the pressure distribution of the hot magnetosphere plasma and transverse currents in the plasma at distances from 8R _E to 12R _E are presented. The data were taken in the vicinity of the equatorial plane onboard the Interball-1 satellite during its passages on October 13, 1995 and March 13, 1996. The pressure was determined from the measurements of particle fluxes by the CORALL, DOK-2, and SKA-2 instruments. The specific features of this experiment made it possible to calculate the pressure with a high accuracy and to determine the distribution of the magnetostatically equilibrium currents in the plasma. It is shown that at the parts of the monotonous increase of the pressure in the earthward direction one can detect regions of plateau in the plasma pressure. A possible origin of the small-scale variations and regions with plateau are discussed. A comparison of the measured pressure profiles with the pressure profiles in the Tsyganenko and Mukai-2003 model is performed. Transverse currents flowing in the plasma are calculated assuming magnetostatic equilibrium.