Stability and Instability of Hierarchical Triple Systems

We computed the dynamical evolution of hierarchical triple stars in which both orbits are initially circular, and determine the lower limit to the ratio of periods (outer/inner) for which there is dynamical stability. We found for some mass ratios resonance-like behaviour that occurs in a limited range of initial period ratio. Some resonances are ‘disruptive’; that is, for a small range of initial period ratio we find that the system is not able to settle down to a quasi-steady hierarchical state, but instead disrupts. However, below as well as above this disruptive range there are considerable ranges of initial period ratio where the hierarchical state appears to be stable, at least for the length of integration time we took which was sometimes as much as 10,000 outer orbits. The mass ratios are identified for which different types of unstable behaviour, such as an escape of the distant body without exchange, many exchanges in the limited space without escape, formation of new long-live hierarchy, or an escape of one body after a few exchanges, occur for ratios of periods slightly below the limit of stability. We discuss the relevance of the above behaviour to observed close triples, the closest of which is λ Tau (period ratio 8.3).

Download Full-text

Fern speciation and biogeography

Proceedings of the Royal Society of Edinburgh Section B Biological Sciences ◽

10.1017/s0269727000008332 ◽

1985 ◽

Vol 86 ◽

pp. 353-360 ◽

Cited By ~ 5

Author(s):

Rolla Tryon

Keyword(s):

New Species ◽

Parental Species ◽

Local Environment ◽

Limited Range ◽

Ecological Adaptation ◽

Species Selection ◽

Potential Range ◽

Small Range ◽

Extensive Range ◽

Selection Of

SynopsisThe most common kinds of speciation result in new species that initially have a small range. These will develop a limited or an extensive range depending upon the geographic extent of the environment to which they are adapted. A significant element in the extent of the potential range of a new species is the adaptation inherited from the parental species. Selection of a parental species for a local environment at one site can lead to a narrow ecological adaptation and often to a limited potential range. These species are likely to produce derived ones that also have a limited range, and these derivates will increase the regional species endemism and diversity. Selection of a parental species for migration to other sites can lead to a broader ecological adaptation and often to a broad potential range. These species are more likely to produce derived ones that also have an extensive range, and these derivates will increase regional species diversity.

Download Full-text

The role of big eddies in homogeneous turbulence

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1949.0007 ◽

1949 ◽

Vol 195 (1043) ◽

pp. 513-532 ◽

Cited By ~ 45

Keyword(s):

Fourier Transforms ◽

Stokes Equations ◽

Initial Period ◽

Rate Of Change ◽

Homogeneous Turbulence ◽

Wave Number ◽

Small Range ◽

Large Wave ◽

Correlation Tensor ◽

Spectrum Function

Recent experimental work on the decay of isotropic turbulence has shown that big eddies play an important part in the motion. There is a range of eddy sizes which, during the initial period of decay, contains a negligible proportion of the total energy and is excluded from the similarity possessed by the smaller eddies. This paper examines the motion associated with this small range of large wave-lengths in the more general case of homogeneous turbulence. For this purpose it is convenient to introduce a spectrum tensor, defined as the three-dimensional Fourier transform of the double-velocity correlation tensor. This spectrum function is also suitable for the application of similarity hypotheses, unlike the conventional one-dimensional spectrum function. The properties of the spectrum as a function of the wave-number vector k, are discussed with particular reference to small values of the magnitude k . When k is small the energy per unit interval of wave-number magnitude varies as k 4 . The rate of change of the spectrum function is obtained from the Navier-Stokes equations in terms of Fourier transforms of the triple-velocity and pressure-velocity mean values. After taking into account the continuity condition it is found that the terms of the first and second degree in the expansion of the spectrum function in powers of components of k are constant throughout the decay. The biggest eddies of the turbulence are therefore permanent, being determined wholly by the initial conditions, and are dominant in the final period when the smaller eddies have decayed. The action of smaller eddies on the invariant big eddies is equivalent to that of a turbulent viscosity, the value of which may vary with direction. The implications of the analysis for similarity hypotheses are discussed briefly.

Download Full-text

Companion-driven evolution of massive stellar binaries

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stz1846 ◽

2019 ◽

Vol 488 (2) ◽

pp. 2480-2492 ◽

Cited By ~ 6

Author(s):

Sanaea C Rose ◽

Smadar Naoz ◽

Aaron M Geller

Keyword(s):

Dynamical Evolution ◽

Dynamical Process ◽

Tidal Dissipation ◽

Triple Systems ◽

Secular Evolution ◽

Gravitational Perturbations ◽

Orbital Configuration ◽

General Relativistic ◽

X Ray Binaries ◽

Insight Into

ABSTRACT At least $70\, {\rm per\, cent}$ of massive OBA-type stars reside in binary or higher order systems. The dynamical evolution of these systems can lend insight into the origins of extreme phenomena such as X-ray binaries and gravitational wave sources. In one such dynamical process, the Eccentric Kozai–Lidov (EKL) mechanism, a third companion star alters the secular evolution of a binary system. For dynamical stability, these triple systems must have a hierarchical configuration. We explore the effects of a distant third companion’s gravitational perturbations on a massive binary’s orbital configuration before significant stellar evolution has taken place (≤10 Myr). We include tidal dissipation and general relativistic precession. With large (38 000 total) Monte Carlo realizations of massive hierarchical triples, we characterize imprints of the birth conditions on the final orbital distributions. Specifically, we find that the final eccentricity distribution over the range of 0.1–0.7 is an excellent indicator of its birth distribution. Furthermore, we find that the period distributions have a similar mapping for wide orbits. Finally, we demonstrate that the observed period distribution for approximately 10-Myr-old massive stars is consistent with EKL evolution.

Download Full-text

Simulations of common envelope evolution in triple systems: circumstellar case

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa3242 ◽

2020 ◽

Vol 500 (2) ◽

pp. 1921-1932

Author(s):

Hila Glanz ◽

Hagai B Perets

Keyword(s):

Binary Systems ◽

Dynamical Evolution ◽

Triple Systems ◽

Common Envelope ◽

The Core ◽

Compact Binary ◽

Transient Electromagnetic ◽

Compact Binary Systems ◽

Binary Mergers ◽

Hydrodynamical Simulations

ABSTRACT The dynamical evolution of triple stellar systems could induce the formation of compact binaries and binary mergers. Common envelope (CE) evolution, which plays a major role in the evolution of compact binary systems, can similarly play a key role in the evolution of triples. Here, we use hydrodynamical simulations coupled with few-body dynamics to provide the first detailed models of the triple common envelope (TCE) evolution. We focus on the circumstellar case, where the envelope of an evolved giant engulfs a compact binary orbiting the giant, which then in-spirals into the core of the evolved star. Through our exploratory modelling, we find several possible outcomes of such TCE: the merger of the binary inside the third star’s envelope; the disruption of the in-spiralling binary following its plunge, leading to a chaotic triple dynamics of the stellar core and the two components of the former disrupted binary. The chaotic evolution typically leads to the in-spiral and merger of at least one of the former binary components with the core, and sometimes to the ejection of the second, or alternatively its further now-binary CE evolution. The in-spiral in TCE leads to overall slower in-spiral, larger mass ejection, and the production of more aspherical remnant, compared with a corresponding binary case of similar masses, due to the energy/momentum extraction from the inner-binary. We expect TCE to play a key role in producing various types of stellar-mergers and unique compact binary systems, and potentially induce transient electromagnetic and gravitational wave sources.

Download Full-text

The Detection of Triple Systems Through the Precession of the Nodes Effect

International Astronomical Union Colloquium ◽

10.1017/s0252921100009921 ◽

1983 ◽

Vol 62 ◽

pp. 120-124 ◽

Cited By ~ 1

Author(s):

T. Mazeh ◽

M. Mayor

Keyword(s):

Orbital Period ◽

Velocity Variation ◽

Mass Center ◽

Period Ratio ◽

Triple Systems ◽

Close Pair ◽

Radial Velocity Variation ◽

A New Technique ◽

A Current ◽

Triple Stars

This paper is a report on a new technique for detecting close triple stars and on a current observational project with some preliminary results, using this method. The goal of the project is to discover new triple systems where the ratio between the long and the short orbital period is small. A system will be called a close triple system if its short orbital period is of several days.In most of the known triple stars (Fekel 1981), the period ratio is very large. This can be attributed to the fact that the wide orbit has been detected visually in most cases. In the few exceptions (e.g. λ Tau, Ebbighausen and Struve 1956), the wide orbit was discovered by detecting additional periodicity in the derived velocity of the close pair mass center γ As this small modulation is imposed on the large radial velocity variation associated with the orbit of the close pair, many triple systems could have escaped this kind of detection. Consequently, the present typical or averaged period ratio might be highly overestimated.

Download Full-text

The Binaries from Unstable Triples. Dynamical Processes of Formation

Highlights of Astronomy ◽

10.1017/s1539299600007644 ◽

1989 ◽

Vol 8 ◽

pp. 143-144

Author(s):

Joanna P. Anosova

Keyword(s):

Computer Simulations ◽

Dynamical Evolution ◽

Close Binary ◽

Close Binaries ◽

State University ◽

Dynamical Processes ◽

Triple Systems ◽

Gravitational Problem ◽

Three Body

The dynamical processes of formation, evolution and disruption of binaries may be effectively studied by computer simulations in the N > 3-body gravitational problem. As a result of analysis of these investigations of diverse authors, the classification of the dynamical processes of formation of wide and close binaries may be proposed (see Table 1). This Table shows the following general processes: I-triple approaches of the single bodies; II-approaches of binaries with single bodies; Ill-escape from physical triples. The actions of these processes, and kinetics of a frequency of binaries in general field were studied at the Astronomical Observatory of the Leningrad State University (1965-1988) by computer simulations in the three-body problem. More than 3.104 orbits with negative total energy E < 0 and 5.104 with E > 0 have been run on the computers. The film “Dynamical evolution of triple systems” was produced. Part I of this movie shows the evolution of the unstable non-hierarchical triplet as well as the processes of formation, evolution, and disruption of temporary wide and final close binaries inside the physical triples. Part II of film presents in detail the trajectories of the bodies on the triple approaches of “fly-by”-and of “exchange”-types. The triple approach of “fly-by”-type results often in an escape from triple as well as the formation of final close binary. The triple approach of “exchange”-type consists as a rule of a few close double approaches of bodies and rarely results in an escape from triplet, it results in formation of temporary wide binary inside triplet. Part III of movie presents the trajectories of the different-mass bodies: an escape of the minimum-mass body, the intermediate-mass body, and the maximum-mass body as well as a formation of binaries with different-mass components.

Download Full-text

Criteria of Dynamical Isolation of Binaries and Multiples

Symposium - International Astronomical Union ◽

10.1017/s0074180900230301 ◽

1996 ◽

Vol 169 ◽

pp. 531-532

Author(s):

Ludmila Kiseleva ◽

Joanna Anosova

Keyword(s):

Strong Influence ◽

Dynamical Behavior ◽

Close Binary ◽

Dynamical Processes ◽

Objective Criterion ◽

Triple Systems ◽

Mass Ratios ◽

Hierarchical Stability ◽

New Criterion

In order to obtain an objective criterion for dynamical isolation of binaries within systems of large multiplicity we study numerically the dynamical behavior and average parameters of stable hierarchical triple systems containing a close binary. Using the new criterion for hierarchical stability of triple systems with different mass ratios of components (Kiseleva, Eggleton, Anosova 1994; Kiseleva, Eggleton, Orlov 1994) the perturbing force from the outer body on the close inner binary is estimated. On this basis, the critical separations are obtained when both inner and outer orbits are practically not perturbed. Because the dispersion of masses has a very strong influence on dynamical processes in N-body systems, mass ratios of subsystems, and sometimes within subsystems, should always be taken into account.

Download Full-text

Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816500146 ◽

2016 ◽

Vol 13 (02) ◽

pp. 1650014 ◽

Cited By ~ 11

Author(s):

Tiberiu Harko ◽

Praiboon Pantaragphong ◽

Sorin V. Sabau

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Differential Equations ◽

Cold Dark Matter ◽

Evolution Equations ◽

Dynamical Evolution ◽

Finsler Spaces ◽

Stability And Instability ◽

The Relationship ◽

Autonomous Dynamical Systems

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a nonlinear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory, the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second-order differential equations. In this paper, we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary [Formula: see text]-dimensional first-order differential equations. We investigate in detail the properties of the [Formula: see text]-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the [Formula: see text]-Cold Dark Matter ([Formula: see text]CDM) cosmological models, respectively.

Download Full-text