The structure of non-hierarchical triple system stability regions

A detailed study of the two-dimensional initial conditions region section in the planar three-body problem is performed. The initial conditions for the three well-known stable periodic orbits (the Schubart’s orbit, the Broucke’s orbit and the eight-like orbit) belong to this section. Continuous stability regions (for the fixed integration interval) generated by these periodic orbits are found. Zones of the quick stability violation are outlined. The analysis of some concrete trajectories coming from various stability regions is performed. In particular, trajectories possessing varying number of “eights” formed by moving triple system components are discovered. Orbits with librations are also found. The new periodic orbit originated from the zone siding with the Schubart’s orbit region is discovered. This orbit has reversibility points (each of the outer bodies possess a reversibility point) and two points of close double approach of the central body to each of the outer bodies. The influence of the numerical integration accuracy on the results is studied. The stability regions structure is preserved during calculations with different values of the precision parameter, numerical integration methods and regularization algorithms of the equations of motion.     

Search for periodic orbits in the general three-body problem

An original method for searching for regions of initial conditions giving rise to close to periodic orbits is proposed in the framework of the general three-body problem with equal masses and zero angular momentum. Until recently, three stable periodic orbits were known: the Schubart orbit for the rectilinear problem, the Broucke orbit for the isosceles problem, and the Moore eight-figure orbit. Recent studies have also found new periodic orbits for this problem. The proposed method minimizes a functional that calculates the sum of squared differences between the initial and current coordinates and the velocities of the bodies. The search is applied to short-period orbits with periodsT < 10τ, where τ is the mean crossing time for the components of the triple system. Twenty one regions of initial conditions, each corresponding to a particular periodic orbit, have been found in the current study. A criterion for the reliability of the results is that the initial conditions for the previously known stable periodic orbits are located inside the regions found. The trajectories of the bodies with the corresponding initial conditions are presented. The dynamics and geometry of the orbits constructed are described.     

Criteria for the stability of triple systems and their application to observations of multiple stars

Criteria for stability of triple systems are studied and compared with the results of numerical simulations obtained for model triple systems and observed multiple stars. The results for the stability analyses using two new criteria—those of Aarseth and of Valtonen et al.—agree with the simulation results in 98% of cases. Thus, these criteria can be used to analyze the stability of systems for which direct modeling of their dynamical evolution is difficult (for example, because not all orbital parameters for their subsystems are known). The last published version of the “Multiple-Star Catalog” of Tokovinin is analyzed to search for systems that may be unstable according to the two new criteria. More detailed studies are carried out for the HD 284419 (T Tau) system. The parameters of the apparent motion method is used to obtain new orbital solutions for this system. The regions of dynamical stability of the system as functions of the orbital parameters are estimated. It is not possible to determine a unique solution for the orbit with the available data; for periods shorter than 300 yr and longer than 5500 yr, the probability of decay of the system on time scales less than 107 yr is high. This conclusion is supported by the application of the stability criteria, as well as direct modeling of the system’s dynamical evolution.     

Analysis of regions of stability in the vicinity of a new periodic S orbit in the three-body problem

The vicinity of a periodic S orbit in the three-body problem with bodies of equal mass is investigated. It is shown that there exists a region of stable motions that preserve the form of the initial periodic orbit. Various sections describing this region are considered. Some specific trajectories from the region of stability and near the boundary of this region are analyzed. Several other regions of stability that are apparently associated with other periodic orbits have also been identified.     

Triple encounters in the linear three-body problem with equal masses

The paper considers triple encounters in the linear three-body problem for the case of equal masses. Triple encounters are described using two parameters: the virial coefficient k and the angle ? such that tan $\varphi = \dot r/\dot \rho$ , where $\dot r$ and $\dot \rho$ are the velocities of the “central” body relative to each of the “outer” bodies. The equations of motion are integrated numerically up to one of the following times: the time for a receding body to turn, the time for this body to reach some critical distance, the time for some escape criterion to be fulfilled, or to some critical time. Evolutionary scenarios for the triple system are determined as a function of the initial conditions. The dependences of the ejection length on k and $\dot \varphi$ are derived. The initial conditions corresponding to escape form a continuous region with k>0.5. The regions into which the right and left bodies depart alternate and are symmetrical about the lines of triple close encounters (?=45°,225°). Regions of stable motions in the vicinity of the central periodic orbit of Schubart (k?0.206; ?=135°,315°) are identified. Linear structures emanate from the peak of the region of stability, which divide the region for the initial conditions into alternating zones with identical evolutionary scenarios.     

An analysis of near-circular lunar mapping orbits

Numerical investigations have been carried out to analyse the evolution of lunar circular orbits and the influence of the higher order harmonics of the lunar gravity field. The aim is to select the appropriate near-circular orbit characteristics, which extend orbit life through passive orbit maintenance. The spherical harmonic terms that make major contributions to the orbital behaviour are identified through many case studies. It is found that for low circular orbits, the 7th and the 9th zonal harmonics have predominant effect in the case of orbits for which the evolution is stable and the life is longer, and also in the case of orbits for which the evolution is unstable and a crash takes place in a short duration. By analysing the contribution of the harmonic terms to the orbit behaviour, the appropriate near-circular orbit characteristics are identified.     

Statistical characteristics of barotropic atmospeheric system and its unstable periodic solutions

This paper is devoted to the problem of approximating an invariant measure and statistical characteristics of barotropic atmospheric model with the help of its periodic trajectories. In this procedure orbits are taken into account according to their weights defined by the orbit instability characteristics. The method comes from the dynamical systems theory where in several specific case (for hyperbolic systems in particular) unstable periodic orbits define the system invariant measure. In our study we show that the system PDF could be reconstructed with the error less than 10% provided that the optimal orbit weight function is chosen.     

Trajectories of bodies in zones of rapidly decaying triple systems

The general three-body problem with equal masses and zero initial velocities is considered. Zones in which the triple systems decay over short times T < 10T _cr are distinguished in the domain of the initial conditions, where T _cr is the mean crossing time for a component of the triple system. These zones form distinct families of structures. Properties of the trajectories of bodies within these structures are described. The structures often display a layered character, with each layer corresponding to triple systems in which a particular body departs during the decay. These layers alternate with zones in which the decay does not occur on such short time scales, and the bodies are flung outward without this leading to a departure, or undergo simple interactions. In the zones of rapid decay, the departure of one of the bodies occurs after one or a few triple encounters between the components.     

Dynamical evolution of multiple stars: Influence of the initial parameters of the system

We have modeled the dynamical evolution of small stellar groups with N=6 components in the framework of the gravitational N-body problem, taking into account possible mergers of stars and ejection of single and binary stars. We study the influence of the initial global parameters of the systems (the mass spectrum, average size, virial factor) on their dynamical evolution. The distribution over states is analyzed for a time equal to 300 initial crossing times of the system. The parameters of binary and stable triple systems that form are studied, as well as the properties of ejected single and binary stars. The rate of dynamical evolution in both expanding and contracting groups is higher than in systems in a state of virial equilibrium. The dynamical evolution is more intense in the case of unequal masses than when the system initially consists of equal-mass stars. In most cases, the evolution of a group ends with the formation of a binary or stable triple system. The semimajor axes of the binaries range from several hundredths to several times the initial size of the system. The distribution of the eccentricities of the binaries formed is consistent with an f(e)=2e law. When the initial size of the group is small, the number of final binaries with large eccentricities, and also of stable triple systems with elongated inner-binary orbits, decreases due to merging. As a rule, stable triple systems are substantially hierarchical (the average ratio of the semimajor axes of the inner and outer binaries is 1: 20). On average, the eccentricities of the inner binaries exceed those of the outer binaries: they are equal to \(\overline {e_{in} } \approx 0.7\) and \(\overline {e_{ex} } \approx 0.5\), respectively. The velocities of ejected stars are from several to several tens of km/s, and tend to increase as the initial size of the system, and hence its virial coefficient, decreases.     

Formation of low-mass X-ray novae with black holes from triple systems

We apply a population synthesis technique to study the formation and evolution of low-mass X-ray binaries with black holes, observed as X-ray novae, from hierarchical triple systems. A scenario is suggested in which an inner close binary system evolves into an X-ray system with a large mass ratio. The high rate of accretion onto the neutron star leads to a common envelope stage, which may result in the formation of a Thorne-Zytkow (TZ) object. During its evolution, the envelope of the TZ object expands, encompassing the third star. The recurrent common-envelope stage decreases the size of the orbit of the third star, leading to the formation of a lowmass X-ray nova with a black hole. The dynamical stability of triple systems automatically ensures that only lowmass X-ray novae form. We also consider the possible formation of an X-ray nova from a binary in the case of asymmetrical core collapse during a supernova explosion.     

Dark energy in the three-body problem: Wide triple galaxies

The structure and evolution of triple galaxy systems in the presence of the cosmic dark-energy background is studied in the framework of the three-body problem. The dynamics of wide triple systems are determinedmainly by the competition between the mutual gravitational forces between the three bodies and the anti-gravity created by the dark-energy background. This problem can be solved via numerical integration of the equations of motion with initial conditions that admit various types of evolutionary behavior of the system. Such dynamical models show that the anti-gravity created by dark energy makes a triple system less tightly bound, thereby facilitating its decay, with a subsequent transition to motion of the bodies away from each other in an accelerating regime with a linear Hubble-law dependence of the velocity on distance. The coefficient of proportionality between the velocity and distance in this asymptotic relation corresponds to the universal value H_Λ = 61 km s^?1 Mpc^?1, which depends only on the dark-energy density. The similarity of this relation to the large-scale recession of galaxies indicates that double and triple galaxies represent elementary dynamical cells realizing the overall behavior of a system dominated by dark energy on their own scale, independent of their masses and dimensions.     

Dynamical evolution of multiple stars

We have modeled the dynamical evolution of small groups of N=3–18 stars in the framework of the gravitational N-body problem, taking into account possible coalescences of stars and the ejection of single and binary stars from the system. The distribution of states is analyzed for a time equal to 300 initial crossing times of the system. The parameters of the binaries and stable triple systems formed, as well as those of ejected single stars, are studied. In most cases, the evolution of the group results in the formation of a binary or stable triple system. The orbital eccentricities of the binaries formed are distributed according to the law f(e)=2e. As a rule, stable triple systems display pronounced hierarchy (the mean ratio of the semimajor axes of the outer and inner binaries is about 20:1). Stars are ejected with velocities from several km/s to several tens of km/s. The results of the modeling are compared with the parameters of observed wide binaries and triple systems.     

The Lyapunov exponents in the dynamics of triple star systems

The dynamics of weakly heirarchical triple stars with equal masses are considered. Full spectra of the Lyapunov exponents are found via numerical integration of the orbits, for various initial configurations of the systemin the planar problem and with initial conditions in the vicinity of the 2 : 1 resonance (i.e., with the initial ratios of the periods of the outer and inner binaries being close to 2 : 1). Dependences between the Lyapunov time and the disruption time of the systemare constructed for initial conditions near and far from resonance. The character of these relationships is different near and far from resonance, corresponding to two kinds of Hamiltonian intermittency. The trajectories “stick” to the regular component in phase space near resonance, while this effect is not dominant far from resonance. Analysis of the distributions of the disruption times of the triple systems for initial conditions near and far from resonance confirm these conclusions.     

Ore-Forming Processes and Dissipative Structures

The theory of dissipative structures is applied in this paper to probing into the dynamics, temporal struc-tures and spatial structures of ore-forming processes and the inherent relationships among them. Areas of oreformation are large dynamic systems in development and evolution. The core of ore formation is the "onset ofore-forming processes". and the crux of it is the "transition from mineralization to ore formation". The theoryof bifurcation and theory of fluctuation make possible the access to the solution of this problem. The multiord-er or successive bifurcation of dissipative structures introduces dynamic geochemical processes into geosciencesand inverses the dynamic evolution and temporal rhythms of ore-forming processes. The localization ofdissipative structures introduces dynamic geochemical fields into geosciences and brings to light the causes andmechanisms of the formation and development of geochemical areas of ore formation (regions and zones of oreformation) and their spatial dynamic patterns.     

Galactic systems against the background of the cosmic vacuum: Structure and evolution of the zero-acceleration surface

The structure and evolution of the zero-acceleration surface around wide triple systems of galaxies are studied in detail. (The zero-acceleration surface is the boundary separating regions in which (i) the Newtonian gravitational attraction of the galactic matter and (ii) the Einsteinian universal repulsion of the cosmic vacuum dominate.) For a typical system, this surface is spherical in shape and several megaparsecs in size, and remains nearly unchanged throughout the lifetime of the system. The concept of a boundary surface can also be extended to systems on the largest possible scales, and its general properties are discussed in relation to clusters, superclusters, and voids.     

Imaging and power generation strategies for chandrayaan-1

The Chandrayaan-1 mission proposes to put a 550 kg lunarcraft into Geostationary Transfer Orbit (GTO) using the Polar Satellite Launch Vehicle (PSLV) which will subsequently be transferred into a 100 km circular lunar polar orbit for imaging purposes. In this paper, we describe certain aspects of mission strategies which will allow optimum power generation and imaging of the lunar surface. The lunar orbit considered is circular and polar and therefore nearly perpendicular to the ecliptic plane. Unlike an Earth orbiting remote sensing satellite, the orbit plane of lunar orbiter is inertially fixed as a consequence of the very small oblateness of the Moon. The Earth rotates around the Sun once a year, resulting in an apparent motion of Sun around this orbit in a year. Two extreme situations can be identified concerning the solar illumination of the lunar orbit, noon/midnight orbit, where the Sun vector is parallel to the spacecraft orbit plane and dawn/dusk orbit, where the Sun vector is perpendicular to the spacecraft orbit plane. This scenario directly affects the solar panel configuration. In case the solar panels are not canted, during the noon/midnight orbit, 100% power is generated, whereas during the dawn/dusk orbit, zero power is generated. Hence for optimum power generation, canting of the panels is essential. Detailed analysis was carried out to fix optimum canting and also determine a strategy to maintain optimum power generation throughout the year. The analysis led to the strategy of 180‡ yaw rotation at noon/midnight orbits and flipping the solar panel by 180‡ at dawn/dusk orbits. This also resulted in the negative pitch face of the lunarcraft to be an anti-sun panel, which is very useful for thermal design, and further to meet cooling requirements of the spectrometers. In principle the Moon’s surface can be imaged in 28 days, because the orbit chosen and the payload swath provide adequate overlap. However, in reality it is not possible to complete the imaging in 28 days due to various mission constraints like maximum duration of imaging allowed keeping in view the SSR sizing and payloads data input rate, time required for downlinking the payload data, data compression requirements and visibility of the lunarcraft for the Bangalore DSN. In each cycle, all the latitudes are swept. Due to the constraints mentioned, only 60‡ latitude arc coverage is possible in each orbit. As Bangalore DSN is the only station, half of the orbits in a day are not available. The longitudinal gaps because of non-visibility are covered in the next cycle by Bangalore DSN. Hence, in the firstprime imaging season, only 25% of the prime imaging zones are covered, and an additional threeprime imaging seasons are required for a full coverage of the Moon in two years. Strategy is also planned to cover X-ray payload coverage considering swath and orbit shift.     

人造卫星轨道要素的计算

本文阐述了人造卫星的轨道计算方法。内容包括轨道的形成、分类与要素以及椭圆轨道与圆轨道的设计计算方法。     

Decay of activity complexes and the formation of coronal holes

Analysis of long-term measurements of solar magnetic fields and the flux of UV radiation from the Sun indicates a cause-effect relationship between activity complexs, their residual magnetic fields, and coronal holes. A comparison of the background magnetic fields of the Sun and the evolution of former activity complexes reveals unipolar magnetic regions that form after the decay of these complexes. The latitude and time evolution of unipolar magnetic regions in solar cycles 21–24 is studied. A North-South asymmetry in solar activity is manifest in the distribution of unipolar regions migrating toward higher latitudes. It is shown that, when residual magnetic fields of the opposite polarity reach the polar regions, this leads to a sign change of the polar magnetic field and a decrease in the area of polar coronal holes, or even their complete disappearance. These interactions can explain the triple sign change of the polar magnetic field of the Sun in cycle 21 and the short-term polarity reversals observed in 2010 and 2011.     

Evolution of Wolf-Rayet stars in binary systems: An analysis of the mass and orbital-eccentricity distributions

We have undertaken a statistical study of the component mass ratios and the orbital eccentricities of WR + O close binary, detached main-sequence (DMS), contact early-type (CE), and semidetached (SD) systems. A comparison of the characteristics of WR + O systems and of DMS, CE, and SD systems has enabled us to draw certain conclusions about the evolutionary paths of WR + O binaries and to demonstrate that up to 90% of all known WR + O binaries formed as a result of mass transfer in massive close O + O binary systems. Since there is a clear correlation between the component masses in SD systems with subgiants, the absence of an anticorrelation between the masses of the WR stars and O stars in WR + O binaries cannot be considered evidence against the formation of WR + O binaries via mass transfer. The spectroscopic transitional orbital period P _tr ^sp corresponding to the transition from nearly circular orbits (e _sp<0.1) to elliptical orbits (e _sp≥0.1) is ～14^d for WR + O systems and ～2^d–3^d for OB + OB systems. The period range in which all WR + O orbits are circular \((1\mathop d\limits_. 6 \leqslant P \leqslant 14^d )\) is close to the range for SD systems with subgiants, \(0\mathop d\limits_. 7 \leqslant P \leqslant 15^d \). The large difference between the P _tr ^sp values for WR + O and OB + OB systems suggests that a mechanism of orbit circularization additional to that for OB + OB systems at the DMS stage (tidal dissipation of the orbital energy due to radiative damping of the dynamical tides) acts in WR + O binaries. It is natural to suggest mass transfer in the parent O + O binaries as this supplementary orbit-circularization mechanism. Since the transitional period between circular and elliptical orbits for close binaries with convective envelopes and ages of 5×10⁹ years is \(P_{tr} = 12\mathop d\limits_. 4\), the orbits of most known SD systems with subgiants had enough time to circularize during the DMS stage, prior to the mass transfer. Thus, for most SD systems, mass transfer plays a secondary role in circularization of their orbits.In many cases, the initial orbital eccentricities of the O + O binary progenitors of WR + O systems are preserved, due to the low viscosity of the O-star envelopes and the short timescale for their nuclear evolution until the primary O star fills its Roche lobe and the mass transfer begins. The mass transfer in the parent O + O systems is short-lived, and the number of orbital cycles during the early mass-transfer stage is relatively low (lower than for the progenitors of SD systems by three or four orders of magnitude). The continued transfer of mass from the less massive to the more massive star after the component masses have become equal leads to the formation of a WR + O system, and the orbit's residual eccentricity increases to the observed value. The increase of the orbital eccentricity is also facilitated by variable radial mass loss via the wind from the WR star in the WR + O system during its motion in the elliptical orbit. The result is that WR + O binaries can have considerable orbital eccentricities, despite their intense mass transfer. For this reason, the presence of appreciable eccentricities among WR + O binaries with large orbital periods cannot be considered firm evidence against mass transfer in the parent O + O binary systems. Only for the WR + O binaries with the longest orbital periods (4 of 35 known systems, or 11 %) can the evolution of the parent O + O binaries occur without filling of the Roche lobe by the primary O star, being governed by radial outflow in the form of the stellar wind and possibly by the LBV phenomenon, as in the case of HD 5980.     

开合构造在不同时空尺度下的地学表现

开合构造是研究地球开合运动及其构造特征,分析开合构造体系的形成机制,探索地球成因和演化的一种新假说。不同时空尺度的开合构造在地学上存在不同的表现,需要从不同角度开展研究。以亚欧非邻区巨型开合构造区、地中海大型开合构造区、东大别中型开合构造区为例,研究了开合构造理论在大地构造分区、地震活动以及现代大地测量结果解释等方面的应用。研究表明:(1)依据开合构造观点可将亚欧非邻区巨型开合构造划分为俄罗斯构造集群、非洲构造集群、北亚构造集群、中亚构造集群、南亚构造集群及东亚构造集群;(2)亚欧非邻区的强地震活动与构造集群间的新生代开合构造转换带关系密切;(3)地中海大型开合构造区内的地震剖面及震源机制揭示了地中海—土耳其—伊朗—阿富汗构造转换带现今构造运动主要以合为主;(4)东大别中型开合构造周边的狮子山、黄梅、麻城等台站的地倾斜和地应变、周边GNSS和流动重力观测结果揭示了该区周边存在时间尺度较短、量级较弱的由“合”向“开”的趋势转变,开合运动是近期诸多观测数据趋势出现准同步性变化的共同机理。