Trajectories of bodies in triple systems with moderate lifetimes

The general three-body problem with equal masses and zero initial velocities is considered. Sets of initial conditions for which the triple systems decay over comparatively short times, 10T _cr < T < 20T _cr, are distinguished (T _cr is the mean crossing time for a component of the system). These sets form distinct families of structures in the domain of the initial conditions. The properties of trajectories of the bodies within some of these structures are described. It is shown that the set of families is no more than countable. A new classification for families of trajectories in decaying triple systems is proposed. Some problems in classifying trajectories in the three-body problem that must be addressed in the future are formulated.     

Transition regions between stable periodic solutions of the general three-body problem

The behavior of triple systems in the transition zones located between regions of stability is studied in the framework of the general three-body problem with equal masses and zero angular momentum. It is well known that there are exist three stable periodic orbits, namely, the Schubart orbit for the rectilinear problem, the Broucke orbit for the isosceles problem, and the Moore eight-figure orbit. In the space of the initial conditions, these orbits are surrounded by sets of points where bounded motions are observed over substantial time intervals. The transition zones between the Schubart and Moore orbits and the Moore and Broucke orbits are studied. It is shown that the boundaries of the regions of stability can be either smooth and sharp or diffuse. Beyond these boundary regions, the dynamical evolution of triple systems results in a distant ejection of one of the components, or the decay of the system. The distribution of the times when the stability is lost is constructed, and obeys a power law for long time intervals. Three stages in the evolution of an unstable triple system are identified.     

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.     

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.     

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.     

Types of motions in the rectilinear three-body problem with unequal masses

The motions in the rectilinear three-body problem are analyzed for the case of unequal masses. At the initial time, the central body is equidistant from the outer bodies. The initial conditions for a fixed ratio of the body masses are determined by two parameters: the virial coefficient of the triple system and the ratio of the differences in the velocities of the central body and the two outer bodies. The domains of the initial conditions corresponding to the escape of one of the outer bodies after some number n={1, 2, 3, 4, 5} of passes of the central body through the barycenter of the triple system are identified. When n≤2, these domains are continuous manifolds. When n≥3, structures consisting of “scattered” points begin to emerge. The changes in the topological properties and in the areas of the continuous domains are determined by the variation of the ratios of the body masses. The domains of the initial conditions for long-lived triple systems are also found, as well as areas within these corresponding to stable systems with bound motions.     

A coordinate form for the Sundman surfaces in the general three-body problem

A new coordinate form for the Sundman surfaces is obtained. Singular points of the surfaces are determined and constructed. The Sundman stability is investigated. The Sundman stability of the motion of the Moon in the general three-body problem of the Sun-Earth-Moon is established.     

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.     

Self-similar focusing of rarefaction waves in relativistic collapsing clouds

The development of an inhomogeneous collapse of an initially homogeneous cloud in pressure equilibrium with an external medium is considered in a general-relativistic treatment. It is shown that a rarefaction wave propagating from the outer boundary toward the center of the cloud forms in the initial stage of the collapse. The rarefaction front moves through the collapsing gas at the sound speed, and separates the cloud material into an inner, uniform core and an outer, inhomogeneous envelope. Our analysis distinguishes two possible collapse regimes, depending on the ratio of the focusing time for the rarefaction wave and the free-fall time for the cloud. In massive clouds, the focusing time for the rarefaction wave exceeds the free-fall time, and the collapse of such clouds inevitably leads to the formation of an event horizon and black hole. In low-mass clouds, the focusing time is less than the free-fall time. After the focusing, the collapse of such clouds is fully inhomogeneous, and can be appreciably slowed by the pressure gradient. The compression of a cloud in a transitional regime separating these two scenarios is studied. In this case, a self-similar collapse regime is realized in the central part of the cloud near the focusing time for the rarefaction wave. The constructed self-similar solution describes both early stages in the compression up to the focusing of the rarefaction wave and later stages with the accretion of gas onto the black hole that is formed. Asymptotic distributions of quantities in the inhomogeneous region at large distances from the rarefaction front and in the accreting envelope are found. The structure of space-time in the vicinity of the black hole that is formed at the center of the cloud as a result of the focusing of the rarefaction wave is discussed.     

Energy conservation in the restricted elliptical three-body problem

The energy conservation law for the elliptical three-body problem is derived using an invariant relation corresponding to the Jacobi integral in the circular problem. The minimum-energy surfaces are constructed, which transform in the case of zero eccentricity into zero-velocity surfaces. Some astronomical applications of the results are considered. In particular, it is shown that Roche-lobe overflow displays pulsational behavior in the elliptical three-body problem.     

Stars with relativistic speeds in the Hills scenario

The dynamical capture of a binary system consisting of a supermassive black hole (SMBH) and an ordinary star in the gravitational field of a central (more massive) SMBH is considered in the three-body problem in the framework of a modified Hills scenario. The results of numerical simulations predict the existence of objects whose spatial speeds are comparable to the speed of light. The conditions for and constraints imposed on the ejection speeds realized in a classical scenario and the modified Hills scenario are analyzed. The star is modeled using an N-body approach, making it possible to treat it as a structured object, enabling estimation of the probability that the object survives when it is ejected with relativistic speed as a function of the mass of the star, the masses of both SMBHs, and the pericenter distance. It is possible that the modern kinematic classification for stars with anomalously high spatial velocities will be augmented with a new class—stars with relativistic speeds.     

Broken Hill — A Precambrian hot spot?

Studies on the variations of metamorphism in the Precambrian granulite facies terrain of Broken Hill has revealed that these high-grade rocks are most frequently found in an elliptical zone known as the Broken Hill Basin. The density distribution of these granulite facies rocks, together with their lithological—structural—geophysical—mineralization relationships suggest local conditions of high-grade metamorphism. Several models to account for these observations are possible including lithological, structural-stratigraphic and mineralization controls. However in the light of current views of plate tectonics, a crustal-spreading hot spot model should also be seriously considered. The association of high-grade metamorphism and Broken Hill type mineralization is thought to be significant and this relationship could be applied in the search for ore deposits.     

The decay of triple systems

Numerical simulations have been carried out in the general three-body problem with equal masses with zero initial velocities, to investigate the distribution of the decay times T based on a representative sample of initial conditions. The distribution has a power-law character on long time scales, f(T) ∝ T ^?α , with α = 1.74. Over small times T < 30T _cr (T _cr is the mean crossing time for a component of the triple system), a series of local maxima separated by about 1.0T _cr is observed in the decay-time distribution. These local peaks correspond to zones of decay after one or a few triple encounters. Figures showing the arrangement of these zones in the domain of the initial conditions are presented.     

Exploration rock geochemistry for Pinnacles-type mineralization at Broken Hill, N.S.W., Australia

Geochemical responses in weathered and oxidized surface metasedimentary rocks associated with stratiform lead-zinc mineralization at Stirling Hill (6 km west of Broken Hill) are compared with the geochemical responses in fresh drill core from an equivalent lithostratigraphic section with stratiform lead-zinc mineralization at the Pinnacles Mine (8 km south of Stirling Hill). Mineralization is interpreted as being volcanic exhalative and it lies within highly metamorphosed (sillimanite grade) rocks of the Willyama Supergroup.Surface rocks were classified into groups by discriminant analysis using drill core data from the Pinnacles Mine as the initial training set. The behaviour of elements in surface rocks varies with the rock group but Zn, Pb, Mn, Fe, and Co are leached from all surface rocks relative to fresh drill core.Nothwithstanding the leaching effects of weathering, common geochemical responses to mineralization have been identified in drill core and surface rocks. Coincident positive anomalies for Zn/Ba and Fe/(Na × Ba) ratios and negative anomalies for Na/(Mn × Ca) ratios uniquely define mineralization in both weathered surface rocks and in fresh drill core.The results demonstrate that the pattern of geochemical responses to Pinnacles-type stratiform volcanic-exhalative mineralization in surface rocks has survived the intensive weathering regime in the Broken Hill region.     

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.     

Development of Methods for Navigational Referencing of Circumlunar Spacecrafts to the Selenocentric Dynamic Coordinate System

Astronomy Reports - The initial form of present-day space optical observations contain considerable geometrical and brightness distortions. This problem can be solved based on geometrical...     

似层状岩质边坡倾倒变形破坏过程数值模拟

在构造裂隙组合切割下形成的反倾向"似层状"结构边坡在工程中普遍存在。文章以浙江某梯级电站厂房后边坡为例,通过运用二维离散元和三维有限差分法,对"似层状"岩质边坡倾倒变形特征及破坏演化历程进行模拟,结果表明这类边坡变形破坏主要经历三个时期:初期为块体间的相互剪切错动和局部缓倾节理的剪切蠕变;中期岩体加剧倾倒后推动坡体前缘滑动,在坡体表面形成倾倒台阶后地表拉裂;后期倾倒变形剪切错动向坡体内部扩展,剪切带相互连通并逐步追踪缓倾结构面或强弱风化分界面,在空间形态上形成具有"台梯状分布"的滑动带,为深层蠕滑创造条件。     

2.5维起伏地表条件下坐标变换法直流电场数值模拟

研究起伏地表对视电阻率分布的影响是进行地形校正的基础。由于很难处理不规则边界,计算简洁且效率高的有限差分方法很少用于解决起伏地表问题。为了解决该问题,引入曲化平思想,实现了一种基于坐标变换法的起伏地表条件下的直流电场数值模拟方法。方法从传统的2.5维基本方程及边界条件出发,通过坐标变换将起伏地表问题转化为水平地表问题,并利用有限差分法求解水平地表问题,最后再通过坐标映射得到起伏地表条件下的电位和视电阻率分布。精度分析及计算实例表明：本方法实现简洁,计算误差主要集中在震源附近,整个计算区域内的平均相对误差为1.39%,计算结果满足起伏地表条件下的电位和视电阻率的分布规律。     

Evidence for multiple Palaeoproterozoic thermal events and magmatism adjacent to the Broken Hill Pb---Zn---Ag orebody, Australia

Thermal events at 1690-1680, 1660-1640 and 1600-1570 Ma have been resolved by SHRIMP U---Pb geochronological study on zircons and monazites from seven localities near to the Broken Hill Pb---Zn---Ag orebody, Australia. The earliest-recognized thermal event included intrusion of now deformed granites such as Rasp Ridge Gneiss and Alma Gneiss and intrusion of gabbro at Round Hill. Previously these have been interpreted as volcanic in origin, and have been assigned to different stratigraphic units of the Palaeoproterozoic Willyama Supergroup. Because these rocks are intrusions, they should be removed from the Supergroup stratigraphic sequence. The 1640–1660 Ma thermal event reached upper amphibolite to granulite conditions and produced melt segregations in parts of the Rasp Ridge Gneiss. Granites of this age are the Purnamoota Road Gneiss, previously correlated with 1690-1680 Ma rocks assigned to the Hores Gneiss stratigraphic unit, and granitic veins within Sundown Group metapelites. The 1600-1570 Ma thermal event also reached upper amphibolite to granulite conditions. The only possible 1600-1570 Ma intrusive rock reported in this study is ‘Lf-leucogneiss’ (granite) at the Purnamoota Road locality. Melt segregations of this age have been found in the Round Hill gabbro and metamorphic segregations have been found in the Purnamoota Road Gneiss. The granite intrusions and segregations are absolute time markers for fabric development and therefore can be used to re-evaluate tectonothermal evolution of rocks close to the Broken Hill Pb---Zn orebody. Within the studied rocks several discrete high grade deformation phases have been observed. The earliest detected deformation is older than 1640–1660 Ma, but syn- or post 1690 Ma. A later deformation phase can be constrained to be pre-or syn 1640–1660 Ma and a yet later deformation phase to be syn- or post- 1600-1570 Ma. The current consensus classifies the Broken Hill Pb---Zn---Ag orebody as the metamorphosed equivalent of classic SEDEX (sedimentary-exhalative) deposits, deposited at ca 1690 Ma. This interpretation heavily relies on the Hores Gneiss being a volcanic marker horizon, because the orebody is situated, apparently conformably, within the Hores Gneiss. However, results of this study show that rocks assigned to the Hores Gneiss are of different age, thus do not form a reliable marker horizon. The present results suggest that in the Thackaringa and Broken Hill Groups in the vicinity of Broken Hill, true supracrustal rocks are ≥ 1690 Ma, rather than ca 1690 Ma as previously suggested. Large parts of rocks surrounding the orebody are intrusions and together with their host supracrustal rocks were metamorphosed and locally remelted at 1660-1640 and 1600-1570 Ma.     

Sensitivity analysis applied to slope stabilization at failure

This article discusses how sensitivity analysis is a sound assessment tool for selecting the most efficient stabilization method of slopes at failure. A discretized form of the variational approach is used not only for performing sensitivity analysis but to locate the critical slip surface, i.e., the sensitivity analysis is carried out in the same way as it is done in optimization problems. This method supplies a robust formulation and methodology for obtaining the sensitivities of the safety factor with respect to both the soil parameters and the slope profile, stating the slope stabilization design as a relatively simple minimization problem. Two well known examples, as the Selset landslide and the Sudbury Hill slip are used to illustrate the application of the method and to highlight both its capabilities and limitations.