Main features of dynamical escape from three-dimensional triple systems

The dynamical evolution of triple systems with equal and unequal-mass components and different initial velocities is studied. It is shown that, in general, the statistical results for the planar and three-dimensional triple systems do not differ significantly. Most (about 85%) of the systems disrupt; the escape of one component occurs after a triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (about 15%) of stable or long-lived systems is formed if the angular momentum is non-zero. Averages, distributions and coefficients of correlations of evolutionary characteristics are presented: the life-time, angular momentum, numbers of wide and close triple approaches of bodies, relative energy of escapers, minimum perimeter during the last triple approach resulting in escape, elements of orbits of the final binary and escaper.     

TRIMOR – three-dimensional correlation technique to analyse multi-order spectra of triple stellar systems: application to HD 188753

This paper presents a new algorithm, trimor , to analyse multi-order spectra of triple systems. The algorithm is an extension of tricor , the three-dimensional correlation technique that derives the radial velocities of triple stellar systems from single-order spectra. The combined correlation derived from many orders enables the detection and the measurement of radial velocities of faint tertiary companions. The paper applied trimor to the already available spectra of HD 188753, a well-known triple system, yielding the radial velocities of the faintest star in the system. This rendered the close pair of the triple system a double-lined spectroscopic binary, which led to a precise mass ratio and an estimate of its inclination. The close-pair inclination is very close to the inclination of the wide orbit, consistent with the assertion that this triple system has a close to coplanar configuration.     

The close triple approach

A new method is described for representing the motion in the planar problem of three bodies when all three point masses simultaneously come close to each other. The main results are (1) that the motion during the critical phase of closest approach is intimately connected with triple parabolic escape and (2) that a sufficiently close triple approach generally leads to the escape of one body witharbitrarily high asymptotic velocity.     

Close triple star systems

Triple star systems, especially those in which one star has very small mass may be more common than has been generally considered. Here is summarized some of the recent evidence supporting this possibility.Communication presented at the International Conference on Astrometric Binaries, held on 13–15 June, 1984, at the Remeis-Sternwarte Bamberg, Germany, to commemorate the 200th anniversary of the birth of Friedrich Wilhelm Bessel (1784–1846)     

Dynamical evolution of triple systems

This article reviews numerical experiments on the three-body problem carried out at the Leningrad University Astronomical Observatory during the past 20 years. Systematic studies of triple systems with negative total energy have yielded the following main results. Most (95%) of the systems decay; the decay always occurs after a close triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (5%) of quasi-stable systems is formed if the angular momentum is non-zero. The qualitative evolution in three-dimensional cases is the same as for planar systems. Small changes in initial conditions sometimes lead to substantial differences in the final outcome. The decay of triple systems is a stochastic process similar to radioactive decay. The estimated mean lifetime is 100 crossing times for equal-mass components and decreases for increasing mass dispersion.A classification of the close triple approaches which lead to immediate escape is given for equal-mass systems as well as for selected sets of unequal components. Detailed studies of close triple approaches by computer simulations reveal that the early evolutions is determined by the initial ratio of the interaction forces. The review concludes by discussing applications of the results to observational problems of stellar and extragalactic systems.     

Dynamics of rotating triple systems

We investigate the dynamical evolution of 100 000 rotating triple systems with equal-mass components. The system rotation is specified by the parameter ω=?c²E, where c and E are the angular momentum and total energy of the triple system, respectively. We consider ω=0.1,1, 2, 4, 6 and study 20 000 triple systems with randomly specified coordinates and velocities of the bodies for each ω. We consider two methods for specifying initial conditions: with and without a hierarchical structure at the beginning of the evolution. The evolution of each system is traced until the escape of one of the bodies or until the critical time equal to 1000 mean system crossing times. For each set of initial conditions, we computed parameters of the final motions: orbital parameters for the final binary and the escaping body. We analyze variations in the statistical characteristics of the distributions of these parameters with ω. The mean disruption time of triple systems and the fraction of the systems that have not been disrupted in 1000 mean crossing times increase with ω. The final binaries become, on average, wider at larger angular momenta. The distribution of their eccentricities does not depend on ω and generally agrees with the theoretical law f(e)=2e. The velocities of the escaping bodies, on average, decrease with increasing angular momentum of the triple system. The fraction of the angles between the escaping-body velocity vector and the triple-system angular momentum close to 90° increases with ω. Escapes in the directions opposite to rotation and prograde motions dominate at small and large angular momenta, respectively. For slowly rotating systems, the angular momentum during their disruption is, on average, evenly divided between the escaping body and the final binary, whereas in rapidly rotating systems, about 80% of the angular momentum is carried away by the escaping component. We compare our numerical simulations with the statistical theory of triple-system disruption.     

The three-dimensional motion of Trojan asteroids

The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian pointsL ₄,L ₅, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed.     

Statistical approach to the theory of circumstellar discs

The equations for the viscous motion of a mixture of gas and dust in a gravitational field are derived from the statistics of particle orbits and radiative processes in a general form which gives the Navier-Stokes equation as a special case. Diffusion, partially elastic collisions and — for larger bodies — the gravitational encounters are included. The results are applied to the evolution of circumstellar discs.     

Periodic orbits in three-dimensional planetary systems

Three-dimensional planetary systems are studied, using the model of the restricted three-body problem for Μ =.001. Families of three-dimensional periodic orbits of relatively low multiplicity are numerically computed at the resonances 3/1, 5/3, 3/5 and 1/3 and their stability is determined. The three-dimensional orbits are found by continuation to the third dimension of the vertical critical orbits of the corresponding planar problem     

Initial conditions and dynamics of triple systems

The dynamical evolution of triple systems with equal-mass components and zero initial velocities is studied. We consider two regions of initial conditions: a regionD of all possible configurations of triples and a circleR. The configurations are distributed uniformly within these regions. The calculations have been carried out until a time when escape or conditional escape (i.e. distant ejection) of one component takes place. The accuracy has been checked by doing time-reversed integration. Types of predictable and non-predictable systems are revealed. Averages for a number of evolution parameters are presented: the life-time, minimum perimeter during the last triple approach resulting in escape, semi-major axis and eccentricity of the final binary, and the smallest separation between the components during the evolution. It is shown that the statistical results for the regionsD andR do not differ significantly for the most part. Our results, which have been obtaned by a three-body regularization method, are in good agreement with previous work based on the RK4 integrator and Sundman's time smoothing.     

Improved equations for eccentricity generation in hierarchical triple systems

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In this paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10⁻³ to 10³. We also present more numerical results for short-term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.     

The binaries and triple systems in an irregular field

The dynamical evolution of theN-body systems (N=6) has been studied by numerical simulations. The double and triple subsystems isolated from other bodies have been revealed. The processes of their formation, evolution, and disruption have been followed. At the initial stage of evolution, a common collapse of the system takes place. After that some fast triple approaches of single bodies are possible. At further stages, some physically connected triple subsystems are formed which are disrupted by the intruders or due to their own dynamical unstability. The dissipation mechanisms formulated by van Albada (1968) and Aarseth (1973) have been confirmed. At the final stage, double or hierarchical triple systems are formed.