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.     

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.     

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.     

Characteristics of motions in the 2D isosceles three-body problem with unequal masses

We analyze the general 2D isosceles three-body problem for various ratios ? of the mass of the central body to the mass of each of the other two bodies. We set the initial conditions using two parameters: the virial coefficient k and the parameter \(\mu = \dot r/\sqrt {\dot r^2 + \dot R^2 }\), where \(\dot r\) is the relative velocity of the two outer bodies and \(\dot R\) is the velocity of the central body relative to the center of mass of the outer bodies. We compare statistical dependences between evolutionary parameters of triple systems with various values of ?, and analyze the k and μ dependences of the number of crossings of the center of mass of the triple system by the central body and the lifetime of the system. We construct the functions R_max(r_max), where r_max and R_max are the maximum achievable distances between the outer bodies, and between the central body and the center of mass of the outer bodies in the triple system. The parameter ? proves to be the most important parameter of the problem, and determines the relationship between the measures of the regular and stochastic trajectories. However, there exist “seeds” of stochasticity, even at small ?～10^?2. The measure of the stochastic orbits increases with ?; when ?≥10, virtually the entire region of the initial conditions corresponds to stochastic trajectories.     

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.     

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.     

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.     

P-T paths and problems of high-temperature polymetamorphism

Many Precambrian granulite-facies metamorphic complexes contain so-called straight gneisses, which are massive rocks with a clearly pronounced blastomylonitic texture, lineation, and gneissosity. These rocks occur exclusively in high-temperature ductile shear zones, which can develop either during the primary exhumation of rock complexes or during the overprinting by high-temperature dynamometamorphism. The main criterion for distinguishing between these two types of straight gneisses is the configuration of their P-T trajectories, which are recorded in the mineral assemblages in these rocks and their host gneisses. Ductile shear zones developed in Archean granulite gneisses simultaneously with their exhumation, and, hence, their P-T trajectories are segments of decompression and/or isobaric cooling paths. Straight gneisses in Proterozoic polymetamorphic complexes commonly compose high-temperature ductile shear zones overprinted on Archean granulite complexes, and the P-T paths of these rocks are Z-shaped. This means that, at a constant pressure in the middle part of the continental crust, the T _min of the older P-T trajectory corresponded to T _max of the younger trajectory, and often T _max–T _min > 100°C. Such ductile shear zones commonly have a strike-slip morphology and can be easily seen in aerial photographs and discerned during structural geological surveying. These zones can overprint older gneisses without any notable thermal effect on the latter. Relations of this type were identified in the granulite complexes of Limpopo in South Africa, Sharyzhalgai in the southwestern Baikal area, and Lapland in the Kola Peninsula. The results of our research propose a solution for the well-known problem of the significant discrepancies between the isotopic ages in high-temperature-high-pressure complexes and the partial or complete distortion of radiogenic isotopic systems under the effect of a newly inflowing metamorphic fluid. The application of geochronologic techniques to these situations is senseless, and only P-T trajectories provide insight into the actual age relations between the discrete tectono-metamorphic stages. It is thus expedient to conduct not only structural studies of metamorphic complexes but also their detailed petrological examination and the calculation of their P-T paths before geochronologic dating.     

Physical and dynamical parameters of the multiple system HD 222326

We present the results of our study of the physical and dynamical parameters of the multiple system HD 222326. A new method for determining the individual radial velocities of components in wide binary and multiple systems in the case of small radial-velocity differences (δV _r ≤ the FWHMfor the line profiles) is suggested and tested for both model systems and the binary HD 10009. This testing yielded the component radial velocities V _{r 1,2} for HD 10009, enabling us to derive the center-of mass velocity, V _γ, for the first time. We determined the radial velocities of the components of HD 222326 from high-resolution spectra, and refined the orbital parameters of the subsystems using speckle-interferometric observations. A combined spectroscopic and speckle interferometric analysis enabled us to find the positions of the components in the spectral type—luminosity diagram and to estimate their masses. It is likely that the components are all in various evolutionary stages after leaving the main sequence. We analyzed the dynamical evolution of the system using numerical modeling in the gravitational three-body problem and the known stability criteria for triple systems. The system is probably stable on time scales of at least 10⁶ years. The presence of a fourth component in the system is also suggested.     

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.     

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.     

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.     

An algorithm for finding composition,molar volume and isochors of CO2-CH4 fluid inclusions from Th and Tfm (for Th < Tfm)

A modified Redlich-Kwong equation of state is used to calculate the solubility of CO₂ in methane at various temperatures and pressures. From the solubility of CO₂ in CH₄ at the triple point and at final melting (T_h < T_fm), and the molar volume of solid CO₂, the volume of solid at the triple point, and the molar volume of the inclusion can be calculated using a mass balance. The pressure at the melting point is calculated from the equation of state.The algorithm predicts composition, molar volume, pressure at final melting and the isochor pressure (for a given temperature of trapping) for CO₂-CH₄ fluid inclusions for the case T_h < T_fm, given T_h, T_fm and experimental data on P_h and d_co2 (solid) at T_h.     

The disruption of close binaries in the gravitational field of a supermassive black hole and the formation of hypervelocity stars

The formation of hypervelocity stars due to the dynamical capture of one component of a closebinary system by the gravitational field of a supermassive black hole (SMBH) is modeled. The mass of the black hole was varied between 10⁶ and 10⁹ M _⊙. In the model, the problem was considered first as a three-body problem (stage I) and then as an N-body problem (stage II). In the first stage, the effect of the inclination of the internal close-binary orbit (the motion of the components about the center of mass of the binary system) relative to the plane of the external orbit (the motion of the close binary around the SMBH) on the velocity with which one of the binary components is ejected was assessed. The initial binary orbits were generated randomly, with 10 000 orbits considered for each external orbit with a fixed pericenter distance r _p . Analysis of the results obtained in the first stage of the modeling enables determination of the binary-orbit orientations that are the most favorable for high-velocity ejection, and estimation of the largest possible ejection velocities V _max. The boundaries of the region of stellar disruption derived from the balance of tidal forces and self-gravitation are discussed using V _max-r _p plots, which generalize the results of the first stage of the modeling. Since a point-mass representation does not enable predictions about the survival of stars during close passages by a SMBH, there is the need for a second stage of the modeling, in which the tidal influence of the SMBH is considered. An approach treating a star like a structured finite object containing N bodies (N = 4000) enables the derivation of more accurate limits for the zone of efficient acceleration of hypervelocity stars and the formulation of conditions for the tidal disruption of stars.     

Resistance of soils to concentrated flow erosion: A review

The soil's resistance to concentrated flow erosion is an important factor for predicting rill and (ephemeral) gully erosion rates. While it is often treated as a calibration parameter in process-based soil erosion models, global change studies require the estimation of erosion resistance from measurable soil properties. Several laboratory and field experiments have been conducted to determine the erosion resistance of various types of soils, but no attempts have been made hitherto to summarize all these data and to explore them for general trends. In this study, all available data on the resistance of topsoils to concentrated flow erosion in terms of channel erodibility (Kc) and critical shear stress (τ_cr) has been collected together with all soil and environmental properties reported in literature to affect the soil erosion resistance. Reported Kc values for cropland topsoils range between 0.002 10^− 3 s m^− 1 and 250 10^− 3 s m^− 1 (n = 470), whereas τ_cr values range between 0 and 15 Pa (n = 522). It is demonstrated that so far, the heterogeneity of measurement methods, the lack of standardized definitions and the shortcomings of the flow shear stress model hamper the comparability of soil erosion resistance values from different datasets. Nevertheless, combining Kc and τ_cr data from different datasets, a general soil erosion resistance ranking for different soil textures can be proposed. The compiled dataset also reveals that tillage practices clearly affect Kc (Kc for conventional tillage > Kc for reduced tillage > Kc for no tillage) but not τ_cr.It was concluded that Kc and τ_cr are not related to each other and that soil and macro-environmental properties affecting the foremost do not necessarily affect the latter as well and vise versa. Often Kc seems to be a more appropriate parameter than τ_cr to represent the differences in soil erosion resistance under various soil and environmental conditions (e.g. bulk density, moisture content, consolidation, tillage). The two parameters represent different quantities and are therefore both needed to characterize the soil's resistance to concentrated flow erosion.     

A method for calculating fractional s-character for bonds of tetrahedral oxyanions in crystals

A method for calculating fractional s-character, f _s , for TO bonds has been devised to apply to TO₄ tetrahedral oxyanions in crystals. These f _s -values rank bond lengths with the better correlations obtained for T atoms associated with larger bond strengths and larger electronegativities. As a simple formula, it is found that 2cot²〈?〉₃ does a good job of estimating f _s where 〈?〉₃ is the triple angle average of the three angles common to a given bond.     

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.     

Metamorphic response in orogens of different obliquity,scale and geometry

Investigation of material flow within transpressional orogens must involve integration of structural and metamorphic datasets. To illustrate the problems in documenting flow vectors we present integrated structural-metamorphic datasets from two transpressional systems; the Kaoko Belt in Namibia and the Kalinjala Shear Zone in South Australia. These orogens experienced widely differing metamorphic responses to transpressional deformation. Integration of kinematic and metamorphic datasets from the Kaoko Belt indicate shallow up-plunging extrusion trajectories in the orogen core, and show that the maximum stretching direction pattern matches the inferred flow vectors. High-grade domains (800–840 °C and 7.0–8.0 kb) in the orogen core developed low-angle upward-verging maximum stretching direction trajectories, whereas a low-grade domain (575–600 °C and 5.0–5.5 kb) in the orogen core has downward-verging lineation trajectories. The barometric differential between these high-grade and low-grade domains is entirely consistent with the angle of plunge of maximum stretching directions within the high-grade domains that were extruded obliquely, for the amount of lateral shear estimated for the orogen core. The Kalinjala Shear Zone in South Australia contrasts strongly with the Kaoko Belt. In this example, the high-grade and high-strain shear zone core of the orogen, experienced high-T/high-P metamorphism with low thermal gradients of 21–26 °C/km and steep decompressive P–T paths. The lower-grade external domains experienced lower-T/lower-P metamorphism with high thermal gradients of 35–37 °C/km. Sub-horizontal maximum stretching directions do not match the vertical extrusional flow in the high-grade core that is indicated by the metamorphic data. This comparison shows that in general and on a gross scale, maximum stretching directions do not necessarily correlate with the real flow vectors experienced during orogenesis. In some cases maximum stretching direction recorded by deformation structures is to some degree decoupled from the vertical component of material flow. Consequently, information pertaining to flow is often partitioned into information derived from deformation structures and information derived from the metamorphic record. These two datasets must be used in concert to obtain realistic constraints on first-order material flow trajectories at orogenic scales. The horizontal component of flow is typically best recorded by structural fabrics (maximum stretching direction and sense of shear), whereas the vertical component is typically best recorded by metamorphic information, such as P–T paths, temperature over depth ratio (G) and metamorphic field gradients (i.e. ΔT, ΔP and ΔG) across the orogen.     

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.     

Bifurcation of 4<Emphasis Type="Italic">π</Emphasis>-periodic solutions of the planar,restricted, elliptical three-body problem

The bifurcation of periodic solutions in the vicinity of triangular libration points of the planar, elliptical, restricted three-body problem is considered. Values µ₀ of the mass parameter µ are determined, such that the problem has non-stationary, periodic solutions close to the libration points for values of µ close to µ₀ and small values of the eccentricity ? ? 0. Asymptotic formulas for these solutions in powers of ? and µ ? µ₀ are determined.