Periodic solutions in three-dimensional restricted three-body problem: 2. Loss of symmetry at the 1:1 resonance

Within the framework of the circular restricted three-body problem a family of inverse periodic orbits around the two attracting bodies (the Egorov’s family) and families generated by it at the 1:1, 2:1, and 3:1 resonances for three-dimensional orbits in the Sun-Earth and Earth-Moon systems are considered. Their relationship with families generated by orbits around the libration points L ₁, L ₂ and L ₃ is investigated. One of the families contains periodic solutions that seem promising as possible orbits for the space radio telescope of the Millimetron project.     

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。     

A new universal equation for the time of transfer between two points of the central gravitational field

The paper presents the derivation of two new equations for calculating the time of transfer between two points of the central gravitational field: for hyperbolic orbits and the universal equation for elliptical and hyperbolic orbits. In the paper we have used as an independent variable, instead of the linear elements (semimajor axis, focal parameter of orbit or a chord connecting the ends of boundary radii of transfer), the angular parameter—the angle between the radius vector of an initial point of transfer and the vector of initial velocity of transfer. Paper’s material is a continuation of that presented in the “Space research“ Journal, vol. 2, March–April, 2009.     

Stable spatial orbits around collinear libration points

A technique of generation of spatial periodic solutions to the restricted circular three-body problem from periodic orbits of the planar problem has been used for the families of orbits around collinear libration points L ₁ and L ₂. Developing the families obtained at the 1: 1 resonance, we have obtained stable solutions both in the Earth-Moon system and in the Sun-Earth system. Of course, the term “around the libration point” is rather conventional; the obtained orbits become more similar to the orbits around the smaller attracting body. The further development of the family of orbits “around” the libration point L ₂ in the Sun-Earth system made it possible to find the orbits satisfying the new, much more rigorous constraints on cooling the spacecraft of the Millimetron project.     

Estimation of accuracy of close encounter performed by the bootstrap method

Ephemerides of the Solar System’s objects are calculated using dynamical models fitted to observations. The accuracy of these ephemerides can be estimated using statistical methods. The methods consist in constructing a set of possible orbits, based on observations. One of these methods, the bootstrap method of repeated sampling, has many advantages: minimal assumptions about distributions of observational errors, easy implementation, etc.     

On the asteroid hazard

The concept of “space patrol” is considered, aimed at discovering and cataloging the majority of celestial bodies that constitute a menace for the Earth [1, 2]. The scheme of “optical barrier” formed by telescopes of the space patrol is analyzed, requirements to the observation system are formulated, and some schemes of sighting the optical barrier region are suggested (for reliable detection of the celestial bodies approaching the Earth and for determination of their orbits). A comparison is made of capabilities of electro-jet engines and traditional chemical engines for arrangement of patrol spacecraft constellation in the Earth’s orbit.     

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.     

Modeling of the aerodynamic moment acting upon a satellite

We consider the issues of modeling the moments of aerodynamic forces acting upon a satellite with gravitational system of stabilization. It is assumed that satellite orbits are almost circular with heights in the range 550–750 km. Simplified analytical expressions are suggested for the aerodynamic moment in the case when a satellite moves in the regime of gravitational orientation. Accuracy of the obtained expressions is estimated to be compared with that of expressions derived under the assumption of constant coefficient of frontal resistance. An analysis is made of short-periodic variations of the atmosphere density occurring due to orbital motion of a satellite. It is demonstrated that these variations can result in a substantial change of the aerodynamic moment, and their approximation by a truncated Fourier series is suggested. Estimates of the accuracy of the suggested approximation are given.     

Novel orbits of Mercury,Venus and Mars enabled using low-thrust propulsion

Exploration of the inner planets of the Solar System is vital to significantly enhance the understanding of the formulation of the Earth and other planets. This paper therefore considers the development of novel orbits of Mars, Mercury and Venus to enhance the opportunities for remote sensing of these planets. Continuous acceleration is used to extend the critical inclination of highly elliptical orbits at each planet and is shown to require modest thrust magnitudes. This paper also presents the extension of existing sun-synchronous orbits around Mars. However, unlike Earth and Mars, natural sun-synchronous orbits do not exist at Mercury or Venus. This research therefore also uses continuous acceleration to enable circular and elliptical sun-synchronous orbits, by ensuring that the orbit's nodal precession rate matches the planets mean orbital rate around the Sun, such that the lighting along the ground-track remains approximately constant over the mission duration. This property is useful both in terms of spacecraft design, due to the constant thermal conditions, and for comparison of images. Considerably high thrust levels are however required to enable these orbits, which are prohibitively high for orbits with inclinations around 90°. These orbits therefore require some development in electric propulsion systems before becoming feasible.     

Orbit lifetime for microparticles in highly elliptical orbits with low perigee

On the basis of numerical experiments, we have shown the principal possibility of long (more than 1 month) and extremely long (more than 1 year) orbit lifetime of technogenic microparticles with radii from 1 to 100 μm injected into the near-Earth space in highly elliptical orbits with low perigee, including the case of an orbit with parameters corresponding to the orbital parameters of the Molniya satellite. Calculations are performed taking into account the perturbing effect on the orbital microparticle motion in the near-Earth space of gravitational perturbation caused by the Earth’s polar oblateness, the solar pressure force (calculated using methods of the Mie theory), and the drag force of neutral component of the background gas under conditions of low, medium, and high levels of solar and geomagnetic activities.     

Radial velocity search for low mass companions of nearby stars

The detection of substellar companions of nearby stars is very difficult, because these objects are very faint and are located close to their bright parent stars. One way to attack this problem is to look for small periodic motion of the bright star, induced by the gravitational attraction exerted by the unseen small companion. Over the past decade stellar spectroscopy has been used by a few teams to monitor the radial velocities of several samples of stars, with a threshold for detection of companions well below the substellar limit of about 0.08 M. So far, only a few possible spectroscopic binaries have been identified where the unseen companions might be substellar.Recently, radio observers discovered that the millisecond pulsar PSR B1257+12 displays periodic variations in the pulse arrival times, indicating the existence of at least three unseen companions in circular orbits around the pulsar. This is the first convincing case for an extra-solar planetary system. This discovery suggests that planetary systems can form in very different situations, and therefore may prove to be common.     

Study of the mode of angular velocity damping for a spacecraft at non-standard situation

Non-standard situation on a spacecraft (Earth’s satellite) is considered, when there are no measurements of the spacecraft’s angular velocity component relative to one of its body axes. Angular velocity measurements are used in controlling spacecraft’s attitude motion by means of flywheels. The arising problem is to study the operation of standard control algorithms in the absence of some necessary measurements. In this work this problem is solved for the algorithm ensuring the damping of spacecraft’s angular velocity. Such a damping is shown to be possible not for all initial conditions of motion. In the general case one of two possible final modes is realized, each described by stable steady-state solutions of the equations of motion. In one of them, the spacecraft’s angular velocity component relative to the axis, for which the measurements are absent, is nonzero. The estimates of the regions of attraction are obtained for these steady-state solutions by numerical calculations. A simple technique is suggested that allows one to eliminate the initial conditions of the angular velocity damping mode from the attraction region of an undesirable solution. Several realizations of this mode that have taken place are reconstructed. This reconstruction was carried out using approximations of telemetry values of the angular velocity components and the total angular momentum of flywheels, obtained at the non-standard situation, by solutions of the equations of spacecraft’s rotational motion.     

Two-craft tether formation relative equilibria about circular orbits and libration points

The relative equilibria of a two spacecraft tether formation connected by line-of-sight elastic forces moving in the context of a restricted two-body system and a circularly restricted three-body system are investigated. For a two spacecraft formation moving in a central gravitational field, a common assumption is that the center of the circular orbit is located at the primary mass and the center of mass of the formation orbits around the primary in a great-circle orbit. The relative equilibrium is called great-circle if the center of mass of the formation moves on the plane with the center of the gravitational field residing on it; otherwise, it is called a nongreat-circle orbit. Previous research shows that nongreat-circle equilibria in low Earth orbits exhibit a deflection of about a degree from the great-circle equilibria when spacecraft with unequal masses are separated by 350 km. This paper studies these equilibria (radial, along-track and orbit-normal in circular Earth orbit and Earth–Moon Libration points) for a range of inter-craft distances and semi-major axes of the formation center of mass. In the context of a two-spacecraft Coulomb formation with separation distances on the order of dozens of meters, this paper shows that the equilibria deflections are negligible (less than 10^?6°) even for very heterogeneous mass distributions. Furthermore, the nongreat-circle equilibria conditions for a two spacecraft tether structure at the Lagrangian libration points are developed.     

圆轨道欠驱动航天器编队重构脉冲控制

针对圆轨道径向或迹向欠驱动航天器编队重构控制问题,提出了欠驱动脉冲控制方法。首先,基于圆轨道欠驱动航天器相对运动动力学模型,分析了两类欠驱动条件下的系统可控性和重构可行性。然后,解析推导了两类欠驱动条件下实现重构所需的最少脉冲次数以及对应的速度增量消耗。最后,设计数值仿真算例,验证了本文提出的欠驱动脉冲控制方法的正确性。仿真结果表明:径向和迹向欠驱动条件下均可实现圆轨道编队重构。与全驱动控制方法相比,欠驱动控制方法可有效避免由推力器故障引起的重构任务失效,故而提高了控制系统的灵活性与可靠性。     

Determination of parameters of rotational motion of the <Emphasis Type="Italic">Monitor-E</Emphasis> spacecraft from the telemetry data on solar batteries’ current

The Monitor-E spacecraft executed uncontrolled flight due to emergency situation, no telemetry information on parameters of the spacecraft’s attitude motion being available. So, the problem arose to determine the spacecraft’s rotational motion from the accessible indirect information—the electric current provided by solar batteries. In this paper the integrated statistical technique is described, that allows one to solve this problem. The values of current, obtained over the time interval some tens of minutes long, have been processed simultaneously by the least squares method using the integration of the equations of spacecraft’s rotational motion. As a result of processing, the initial conditions of motion were estimated, and the spacecraft’s moments of inertia were updated, as well as the angles, specifying solar batteries position in the spacecraft-fixed coordinate system. The results of processing of 12 data sets are presented, which allowed us to reconstruct the actual rotational motion of the spacecraft.     

A Retrospective Geometric Analysis of the Long-Term Evolution of Orbits and the Ballistic Lifetime for Satellites of the Prognoz Series

We consider the applied aspects of the geometrical analysis of solutions in a restricted circular double-averaged three-body problem that are concerned with the design of high-apogee satellite orbits. Based on the analysis of the long-term evolution and the ballistic lifetime for orbit families of the Prognoz satellites launched into their orbits in the period 1972–1995, we suggest some practical suggestions for choosing long-lived high-apogee orbits with taking into account various requirements for the domain of evolution of the orbital elements.     

椭圆轨道编队飞行的典型模态与构型保持控制方法

基于T H方程推导了椭圆参考轨道编队飞行的周期性条件,讨论了椭圆参考轨道卫星编队飞行的典型模态。该模态是圆形参考轨道空间圆形模态的推广。论述了二阶带谐项(J2项)摄动对编队飞行构型保持的影响,基于相平面法提出了一种编队飞行构型保持控制方法。该控制方法不是消极的抵消干扰的影响,而是积极的利用干扰的作用达到节约燃料并精确保持构型的目的。仿真表明,采用该控制方法可对椭圆参考轨道卫星编队构型进行有效的保持。     

Optimization of Bi-Elliptic transfer with plane change—Part I

We introduce a more advanced analysis than that of Roth concerning the following two points of view: (1) We attempt to find an analytical solution of the optimization problem instead of Roth's iterative solution. That means a stress is operated on the analytical approach. (2) We consider the terminal orbits to be elliptic and not circular as Roth assumed. This is a generalization of the problem. We shall complete the solution of this Bi-Elliptic transfer with plane change problem in the second part of the present research paper (I).