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.     

Resonances in the three-body problem with equal masses

Numerical simulations of near-resonant motions occurring in planar rotating systems of three bodies with equal masses are presented for the cases of prograde and retrograde motions of the inner and outer pairs. The ratios n: m of the initial periods of these pairs are within 1 ≤ m ≤ n ≤ 7. It is shown that, as a rule, retrograde are more stable than prograde motions. The k loop structures oriented inwards for prograde and outwards for retrograde motions are revealed. Empirical relations between k and n are found for the main resonances n: 1 and n: 2.     

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.     

Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses

The three body problem with variable masses with two of the bodies being protoplanets is analyzed. The protoplanetary masses are assumed to be much less than the protosolar mass: m ₁(t) ? m ₀(t), m ₂(t) ? m ₀(t). The variations of the body masses over time are assumed to be known. The masses vary isotropically with different rates: ? ₀/m ₀≠? ₁/m ₁, ? ₀/m ₀≠? ₂/m ₂, ? ₁/m ₁≠? ₂/m ₂. The bodies are treated like material points. The problem is described by analogy to the second system of Poincaré elements, based on the equations of motion in a Jacobian coordinate system. Individual aperiodic motions in a quasi-conical cross section are used as the initial, unperturbed, intermediate motions. The expression for the perturbing function does not include terms proportional to third and higher powers of the small masses m ₁ and m ₂. A new analytical expression for the perturbing function analogous to the second system of the Poincaré variables is obtained in the formulation considered using a classical scheme. The analogs of the eccentricities e ₁ and e ₂ and the orbital inclinations i ₁ and i ₂ are considered to be small. The perturbing function accurate to within terms of second order in the small quantities e ₁, e ₂, i ₁, and i ₂ is calculated in a symbolic form using Mathematica package. The equations for the secular perturbations in this protoplanetary three-body problem, with the bodies treated as points with masses varying isotropically with different rates, are obtained. General rigorous analytical solutions to these equations for the secular perturbations describing the evolution of the orbital planes are derived for oblique elements, for arbitrary mass-variation laws. An analog of the Laplace theorem is proved for the orbital inclinations. Analytical formulas are obtained for the temporal variation of the longitudes of the ascending nodes and the inclinations for arbitrary mass variations with different rates. It is shown that the Jacobian node theorem, which is valid in the classical three-body problem with constantmasses, is violated in this problem, unless special initial conditions apply.     

Periodic orbits in the free-fall three-body problem

Periodic solutions of the general free-fall three-body problem are investigated for the case of equal masses. The initial conditions are chosen on a Hill surface in form space. The use of the form space reduces the dimension of the problem and makes it possible to represent the region of possible initial conditions on the Hill surface, together with a color scale. The regions obtained can be used to improve the precision of the initial conditions for the periodic orbits in the free-fall three-body problem.     

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.     

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.     

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.     

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.     

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.     

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.     

青海省治多县扎那日根岩体特征及构造意义

扎那日根岩体主要由早期的黑云母石英闪长岩和晚期的花岗闪长岩组成。花岗闪长岩中单颗粒锆石U-Pb年龄值为216±2Ma和217±7Ma(晚三叠世)。岩石富碱贫钛,属钙碱性岩系。微量元素特征表现为:富集大离子亲石元素K、Ba、Rb及高场强元素Th,相对于其它相邻元素而言,Sr、Nb、Ta略显亏损,Ti、P较强亏损,(87Sr/86Sr)i为0.70403~0.70463,εNd(t)为2.7~3.4。构造背景为火山弧花岗岩,成岩物质源于壳幔混合产物。     

三维应力作用下岩体单个裂隙的渗流特性分析

天然岩体中存在着大量的孔隙和裂隙,这些缺陷不仅改变了岩体的力学性质,也严重影响了岩体的渗流特性。在对现有裂隙岩体渗流特性研究成果进行分析的基础上,讨论了岩体单个裂隙的力学性质和渗流对单个裂隙岩体产生的力学作用,研究了岩体单裂隙渗透系数与岩体三维应力的关系,考虑了裂隙粗糙度对渗流的影响以及不同方向应力对渗透性影响的差异,分析了单个裂隙岩体在三向应力作用下的渗流特征,得出了裂隙所受三维应力与渗透系数关系式,认为垂直于裂隙面的应力对岩体渗透性起主导作用,岩体渗透系数随垂直裂隙面应力的增加而迅速减小。通过与渗流规律试验结果对比分析,证明了所得单个裂隙岩体渗透系数表达式的正确性。     

A 2D,fully-coupled,hydro-mechanical,FDEM formulation for modelling fracturing processes in discontinuous,porous rock masses

This paper presents a new, fully-coupled, hydro-mechanical (HM) formulation for a finite-discrete element method computer code. In the newly-developed, hydraulic solver, fluid flow is assumed to occur through the same triangular mesh used for the mechanical calculations. The flow of a viscous, compressible fluid is explicitly solved based on a cubic law approximation. The implementation is verified against closed-form solutions for several flow problems. The approach is then applied to a field-scale simulation of fluid injection in a jointed, porous rock mass. Results show that the proposed method can be used to obtain unique geomechanical insights into coupled HM phenomena.     

A mixed finite element method for the solution of the magnetostatic problem with highly discontinuous coefficients in 3D

This paper present a new a mixed finite element method for the simulation of magnetostatic problems with highly discontinuous permeability. The method is derived from the well studied mixed formulation for the div-grad system that is known to be accurate for very large discontinuities. The method robustness is demonstrated on a test model problem.     

Canonical variate analysis with unequal covariance matrices: Generalizations of the usual solution

Canonical variate analysis is extended for use when the covariance matrices are not equal. Linear combinations of variates are derived by generalizing either a weighted between-groups approach or the likelihood-ratio test and the associated noncentrality matrix. The usual solution and the two generalizations are compared via generated data for a few typical configurations of means in a situation in which the covariance matrices are in fact equal. The MSE of the canonical variate coefficients and group means for the generalizations are approximately three times those for the usual solution, due to corresponding changes in the variances. Two examples are discussed.     

Microplate motions in the hotspot reference frame

We have computed motions of the major plates (seven large plates and seven medium‐sized plates) and 38 microplates relative to the hotspot reference frame, and present velocities of these 52 plates. Moreover, using updated plate boundaries for the present, we have computed new geometrical factors for plates and microplates, useful for kinematic calculations and to obtain the net‐rotation of the lithosphere and plate velocities in the mean‐lithosphere reference frame. Instead of a continuum or gradational distribution of the plates by size, the plates clearly partition into three groups each having their own characteristics. For the seven large plates, rotation poles generally lie in high latitudes; the seven medium‐sized plates have rotation poles in a restricted equatorial area; the 38 small plates show the greatest scatter. Moreover subsets of the 52 plates reveal differing fractal behaviour: the large, middle and small groupings each have a characteristic fractal dimension, suggestive of microplate clustering. The highest angular velocities occur for some of the smallest plates, with the location of their rotation poles closeby. Terra Nova, 18, 276–281, 2006     

Forecasting the polar motions of the deformable Earth

A mathematical model for the complicated phenomenon of the polar oscillations of the deformable Earth that adequately describes the astrometric data of the International Earth Rotation Service is constructed using celestial mechanics and asymptotic techniques. This model enables us to describe the observed phenomena (free nutation, annual oscillations, and trends) simply and with statistical reliability. The model contains a small number of parameters determined via a least-squares solution using well-known basis functions. Interpolations of the polar trajectory for intervals of 6 and 12 yrs and forecasts for 1–3 yrs are obtained using the theoretical curve. The calculated coordinates demonstrate a higher accuracy than those known earlier.     

The motions of galaxies in the field of the cosmic vacuum

The general solution and general integral of the equations of motion in the field of the cosmic vacuum are constructed. It is shown that the resulting motions of galaxies are along either hyperbolic or rectilinear paths. The laws of motion of galaxies in the field of the cosmic vacuum are formulated. Various forms of the Hubble law are considered. A strict adherence to the Hubble law is not possible for most initial conditions in the sense of the Lebesgue measure. Therefore, it becomes meaningless to search for explanations to deviations from the Hubble law due to any physical factor, apart from the repulsive force of the cosmic vacuum. Phase portraits for the galaxy motions are constructed. It is shown that the Hubble constant should be determined observationally using the most distant galaxies, since the accuracy of the result will be reduced otherwise.     

Factors Affecting the Behaviour of Retaining Structures with Prestressed Anchorages Under 2D and 3D Conditions

Some basic factors which influence the behaviour of the anchored retaining structures are investigated and 2D analyses with three dimensional ones are compared. The parametric analysis under 2D conditions focuses on the influence of significant factors such as the normalized embedded depth of the wall, the resistance of the anchors, the free tendon length, the earth pressure coefficient at rest and the flexibility of the system, on the behaviour of the retaining structure. The analyses are performed in two groups for two representative soil types named S₁ (weathered weak rock) and S₂ (very stiff clay). Qualitative as well as quantitative conclusions are derived from the analyses concerning the influence of the above mentioned factors, mainly on the safety factor as well as on the displacements. The 3D analyses are carried out for various ratios L/H, where L is the length and H the height of the slope. Several trends are confirmed from the results concerning the influence of the above-mentioned factors. It is concluded that the triaxal conditions strongly affect the safety factor, as well as the distribution of horizontal and vertical displacements, mainly in the cases where the ratios L/H ≤ 2. Some serious deviations between 2D and 3D analyses are pointed out and the physical explanation of these differences is given for several cases.