Quasi-synchronous orbits and their employment for the approach of a spacecraft to Phobos

The problem of construction of quasi-synchronous orbits which pass through a prescribed point over the surface of Phobos at a prescribed instant of time is considered. The orbits should pass as close as possible to the surface of Phobos at each passage above the planned region of landing.     

Collisions of stars by oscillating orbits of the second kind

Ordinary estimations of the number of star collisions in our galaxy—by simple kinematic considerations—lead to a very small number of such collisions: about one or even less every millions of years. However star collisions can occur through the following indirect way which has a much higher probability. (a) Binary stars are very common in our galaxy, about 30–50% of the stars. (b) If two binary stars meet a triple system can be formed by an ordinary exchange type motion. (c) A triple system is generally decomposed into the “inner orbit” (i.e. the relative orbit of the two nearest stars) and the “outer orbit” (i.e. the relative orbit of the third star with respect to the center of mass of the two nearest stars). The major axes of these two orbits have generally small perturbations and it is the same for the eccentricity of the outer orbit. On the contrary, if the relative inclination of the two orbits is large, the perturbations of the eccentricity of the inner orbit are important and can even in some cases lead to an eccentricity equal to one, that is to a collision of the two stars of the inner orbit.Such orbits can be called “oscillating orbits of the second kind”, indeed the first oscillating orbits—conceived by Khilmi and described for the first time in an example by Sitnikov—have unbounded mutual distances r_ij, but the system always come back to small sizes, it has an infinite number of very large expansions followed by strong contractions and, in the three-body case, an upper bound of lim inf (r_1.2 + r_1.3 + r_2.3) can be given in terms of the three masses and the integrals of motion. For the oscillating orbits of the second kind the mutual distances r_ij are bounded, but the velocities are unbounded (i.e. lim inf r_ij = 0 for at least one r_ij) and the system goes to a collision if the bodies have non-zero radius even small. The analytical study of the oscillating orbits of the second kind is a part of the general analytical study of the three-body problem, a part which must be valid for large eccentricities and large inclinations. The use of Delaunay's variables and of a Von Zeipel transformation lead to a first order integrable approximation, valid for any eccentricities and any inclinations, and giving the following results: (a) The oscillating orbits of the second kind occur when the angular momentum of the outer orbit has a modulus sufficiently close to the modulus of the total angular momentum of the three-body system. Hence these orbits occur for inclinations in the vicinity of 90°. (b) The oscillating orbits represent a set of positive measure of phase space and the first order study allows to give a rough estimation of the probability of collisions—even for stars of infinitely small radius. This probability, for given initial major axes and eccentricities and for isotropic arbitrary initial orientations, is generally of the order of m₃$\text{R}\text{M}$ (m₃ being the mass of the outer star, M the total mass and R the ratio of the period of the inner orbit to that of outer orbit).One question remains to be solved: how many collisions of stars are due to that phenomenon? That question is difficult because the probability of formation of a triple system by a random meeting of two binaries is very uneasy to estimate. However it seems that, compared to the usual evaluations based on pure kinematic considerations without gravitational effects, the number of collisions must be multiplied by a factor between one thousand and one million.     

关于旋进椭圆的讨论

根据Bertrand定理关于轨道闭合的条件,我们知道当中心引力的形式为径向距离的幂次型函数时,并不总能导致闭合轨道,即运动的质点不一定能返回到它自己的轨迹上去。这个定理指出:一切运动质点其初始条件稍微偏离圆轨道的要求条件时,只有当引力满足平方反比律(即牛顿万有引力定律)或线性定律(即虎克定律)时,轨道才是闭合的。 本文试图采用分析力学中作用角变量的方法,对一般情形下的轨道闭合条件进行讨论。这里我们看到使初始轨道接近于圆轨道的假设是不必要的,而且力的类型可推广到包括平方反比和其它幂次型的力。我们证明了轨道闭合的判别公式,并得到在上述情形下轨道非闭合的结论。 最后,如果中心引力包括有r~(-2)项和r~(-3)项时,象相对论性改正后所得到的那样,轨道就不再是闭合的,而是一旋进椭圆形轨道,椭圆的旋进角速度可以容易地计算出来。     

地球内辐射带核心区环境特征分析及质子屏蔽的蒙特卡罗模拟

辐射带粒子是近地空间卫星总剂量辐射的主要来源。文章分析了内辐射带不同高度轨道的辐射环境特性;并利用Geant4程序,针对内辐射带质子环境进行不同材料的屏蔽效能计算。结果表明:虽然传统的低?高?低原子序数材料三明治屏蔽结构对电子具有较高的屏蔽效能,却并不适用于以质子环境为主的轨道;对于工作在3000 km圆轨道、5年寿命的卫星,若要将总剂量降至30 krad(Si)以下,使用PE屏蔽材料可比Al屏蔽减重28%。     

Pressing Issues of the Day in Studying Deep Space

Two problems in studying deep space are discussed that are, in the author's opinion, the most important. The first is soil sampling from the smaller bodies of the Solar System, such as the Martian satellite Phobos and asteroids of groups C and S of the Main Asteroid Belt. This soil (so-called primordial substance) can help to elucidate some problems of the Solar System's formation; in particular, to construct a reliable model of the internal structure of the Earth. The second problem is to reveal all sufficiently large asteroids penetrating inside the Earth's orbit and to catalog those asteroids that are hazardous from the viewpoint of collision with the Earth. To this end, it is suggested to launch five or six Earth-orbiting spacecraft with telescopes capable of recording objects down to a brightness of 22– 25 ^m . It is pointed out that both problems can be solved in the near future using comparatively cheap standardized space vehicles launched into near-Earth orbits by the Soyuz carrier rocket and boosted further by electro-jet engines of small thrust.     

Interaction of Phobos with the Solar Wind Plasma

The interaction of Phobos with the solar wind is considered both theoretically and using the experimental data of the FGMM magnetometer that were obtained in the course of the Phobos-2mission. It is demonstrated that the ions serving as a source of excitation of magnetosonic waves can be accumulated around Phobos. Examination of the magnetometer data has shown that the observed effects of a local decrease of the magnetic field near the Phobos orbit correspond to magnetosonic waves. Observation of these effects depends on the geometry of an experiment.     

初值不确定轨道可达区域计算

为计算航天器轨道初值不确定造成的可达区域（RD）,提出一种适用于椭圆参考轨道的计算方法。以航天器的名义轨道作为基线,定义一个随时间变化的移动参考平面,在此移动平面上建立名义轨道到可达区域上任一点的距离函数,函数的极大值点即为可达区包络的边界点,从而将问题转换为非线性方程组的求解问题,并对动力学模型部分线性化带来的误差进行了分析。随后采用变参数的自适应同伦算法对方程组进行求解。最后对航天器释放微小卫星的实例进行数值仿真,结果显示该方法可行有效。     

Design of satellite constellations with continuous coverage on elliptic orbits of <Emphasis Type="Italic">Molniya</Emphasis> type

New methods of choosing the structures of satellite constellations (SC) on elliptical orbits of the Molniya type are presented. The methods, using critical inclination and putting the orbit apogee in the Earth’s hemisphere with an area of continuous coverage, are based on geometrical analysis of two-dimensional representation of the coverage conditions and SC motion in the space of inertial longitude of the orbit ascending node and time. The coverage conditions are represented in the form of a certain region. Dynamics of all satellites in this space is represented by uniform motion along a straight line approximately parallel to the ordinate axis, while the satellite system forms a grid. The problem of choosing a minimal (as far as the number of satellites is concerned) SC configuration can be formulated as a search for the most sparse grid. The contemporary advanced methods of computational geometry serve as an algorithmic basis for the problem solution. Design of SC for continuous coverage of latitude belts with the use of kinematically regular systems is considered. A method of analyzing single-track systems for continuous coverage of arbitrary geographic regions is described, which makes a region at any time instant observable by at least one satellite of the system. As an example, SC on elliptical orbits are considered with periods of ～4, 12, and 24 hours.     

从地球到日-火系平动点轨道的转移

平动点轨道特殊的空间位置及动力学特征,使其在深空探测中具有重要的应用。以日-火系平动点轨道(Lissajous与Halo轨道)任务为目标,结合平动点轨道的不变流形理论,研究了小推力转移问题。首先给出了圆型限制性三体动力学模型下平动点附近不变流形(稳定和不稳定流形)高阶分析解以及相应的计算实例。接着以流形分析解为基础,建立了初始小推力轨道优化模型,并利用改进的协作进化算法求解初始小推力轨道。最后将初始轨道离散,采用多点打靶法将最优控制问题转化为参数优化问题,并用序列二次规划方法(SQP)求解。仿真结果证明轨道设计方法的有效性。     

Spacecraft Injection into a Heliocentric Terrestrial Orbit

We consider the problem of injection of a spacecraft into the heliocentric Earth's orbit ahead and/or behind the Earth by 60° and 120° in heliographic longitude. The range of solar and astrophysical problems for which these orbits are necessary is reviewed. The variants of injection into heliocentric orbits work from a low around-Earth orbit with one turn-on of the engine in this orbit and one turn-on at the end of the injection trajectory. In this case, it turns out to be more profitable to put spacecraft into orbit for three or even four revolutions of the Earth about the Sun. The velocities necessary for the start from a low around-Earth orbit, the velocities at the final point of injection, and the fuel mass (relative to the spacecraft mass) necessary for injection are estimated. The problems for which injection to similar orbits is executed, using the low-thrust engine and with a combined regime of injection, are also considered.     

地-月系平动点及Halo轨道的应用研究

地-月系统的平动点L1点及L2点的Halo轨道在探月工程中有重要的应用价值，可分别用于地月连续通信覆盖和月球背面的探测。由于在地-月系统中太阳的引力不可忽略，特别是在长时间作用以后，其动力学行为与摄动力较小的日-地系统有明显的不同。本文分析了如何利用太阳引力进入地-月系统的L1点及L2点的Halo轨道、以及由Halo轨道进入近月轨道的问题，两者综合起来构成了一条完整的地月低能转移轨道。研究结果对探月轨道设计有一定的参考价值。     

Halo orbits in the vicinity of the L 2 libration point in the Sun-Earth system

Methods are proposed for constructing the orbits of spacecraft remaining for long periods of time in the vicinity of the L ₂ libration point in the Sun-Earth system (so-called halo orbits), and the trajectories of uncontrolled flights from low near-Earth orbits to halo orbits. Halo orbits and flight trajectories are constructed in two stages: A suitable solution to a circular restricted three-body problem is first constructed and then transformed into the solution for a restricted four-body problem in view of the real motions of the Sun, Earth, and Moon. For a halo orbit, its prototype in the first stage is a combination of a periodic Lyapunov solution in the vicinity of the L ₂ point and lying in the plane of large-body motion, with the solution for the linear second-order system describing small deviations of the spacecraft from this plane along the periodic solution. The desired orbit is found as the solution to the three-body problem best approximating the prototype in the mean square. The constructed orbit serves as a similar prototype in the second stage. In both stages, the approximating solution is constructed by continuation along a parameter that is the length of the approximation interval. Flight trajectories are constructed in a similar manner. The prototype orbit in the first stage is a combination of a solution lying in the plane of large-body motion and a solution for a linear second-order system describing small deviations of the spacecraft from this plane. The planar solution begins near the Earth and over time tends toward the Lyapunov solution existing in the vicinity of the L ₂ point. The initial conditions of both prototypes and the approximating solutions correspond to the spacecraft’s departure from a low near-Earth orbit at a given distance, perigee, and inclination.     

Bifurcation Regime of a Synchronous Resonance in the Translational–Rotational Motion of Nonspherical Natural Satellites of Planets

Period-doubling bifurcations of the synchronous spin-orbit resonance in the motion of a nonspherical natural planetary satellite along the elliptic orbit are studied. The satellite spin axis is assumed to coincide with the axis of its largest principal moment of inertia and is perpendicular to the orbital plane. The period-doubling bifurcations take place when the value of satellite's dynamical asymmetry parameter falls in the parametric resonance domain. Theoretical dependences of the amplitude of the bifurcation oscillations of a satellite at the pericenter of its orbit upon the eccentricity and dynamical asymmetry parameter are investigated. Three different methods of calculating the amplitude of bifurcation oscillations are presented and compared. These theoretical estimates can be used to predict the opportunity to observe the bifurcation regime. The possibility of the occurrence of the bifurcation regime in the motion of natural planetary satellites is studied. It is concluded that the bifurcation regime is possible in the motion of Deimos, Epimetheus, Helen, Pandora, and Phobos. Phobos is the most probable candidate for finding the bifurcation regime of a synchronous rotation. The identification of such a regime would allow one to impose stringent constraints on the values of the inertial parameters of the satellite observed.     

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。     

On the Precession of Saturn

The precession of Saturn under the effect of the gravity of the Sun, Jupiter and planet’s satellites has been investigated. Saturn is considered to be an axisymmetric (A = B) solid body close to the dynamically spherical one. The orbits of Saturn and Jupiter are considered to be Keplerian ellipses in the inertial coordinate system. It has been shown that the entire set of small parameters of the problem can be reduced to two independent parameters. The averaged Hamiltonian function of the problem and the integrals of evolutionary equations are obtained disregarding the effect of satellites. Using the small parameter method, the expressions for the precession frequency and the nutation angle of the planet’s axis of rotation caused by the gravity of the Sun and Jupiter are obtained. Considering the planet with satellites as a whole preceding around the normal to the unmovable plane of Saturn’s orbit, the satellites effect on the Saturn rotation is taken into account via the corrections in the formula for the undisturbed precession frequency. The satellites are shown to have no effect on the nutation angle (in the framework of the accepted model), and the disturbances from Jupiter to make the main contribution to the nutation angle evolution. The effect of Jupiter on the nutation angle and the precession period is described with regard to the attraction of satellites.     

Elegant propositions of the boundary-value problem

The fundamental ellipse of the two-body, two-point orbital boundary-value problem of orbital mechanics is introduced and its key role in the basic geometry of this famous problem is developed. Analytic formulas for the orbital parameters of conjugate orbits are derived together with further interesting properties of the mean point-locus.     

三角平动点附近高阶解在轨道位置保持中的应用

首先给出三角平动点附近的高阶解析解,并计算了三种特殊的运动类型。以日–地+月系三角平动点附近无长周期运动分量的拟周期轨道作为目标轨道,探讨轨道保持问题。针对三角平动点任务的轨道保持问题,我们研究了两种轨道保持策略,分别为多点打靶轨道保持与重构目标轨道的策略。计算中,将轨道控制问题转化为非线性规划问题,并以优化方法求解。仿真表明优化方法在轨道保持问题求解方面非常有效。     

On the features of long-term evolution of a high-apogee orbit of the SPECTR-R spacecraft

The practical tasks related to qualitative investigation of long-term evolution of high-apogee orbits of artificial Earth satellites (AES), for which the main perturbing factors are gravitational perturbations from the Moon and the Sun, are considered. Attention is given to the problem of the ballistic lifetime of similar orbits, and the issues associated with possibilities of the correction of orbits for ensuring the required duration of their ballistic lifetime are considered. The orbit of the SPECTR-R spacecraft launched in July of 2011 is considered as an example.     

环月飞行器返回轨道设计与优化

针对传统月地返回轨道设计方法在从初始轨道直接转移的局限性,提出一种基于Lambert问题的月地转移轨道设计方法。该方法重新选取了轨道约束参数,并在笛卡尔坐标系下建立了返回轨道动力学模型,并基于该模型全面分析了返回轨道关于约束参数的变化特性。此外,针对带有约束条件的返回轨道优化设计问题,提出了快速收敛的函数构造法,并给出了有效的目标函数形式。以某一在轨环月飞行器的轨道参数作为初始轨道进行返回轨道计算,结果验证了所提出返回轨道设计方法以及函数构造法的有效性。