Analysis of Families of Triple Close Approaches Occurring inStellar Systems in Three Dimensions (II)

This is the second paper of a trilogy dealing with the role of triple encounters with low initial velocities and equal masses in the evolution of stellar systems in three dimensional space. It shows how a condition of complete collapse may be perturbed to obtain well-established families of asymmetric triple close approaches with systematic regularity of escape with the formation of a binary. The main result is that when perturbation is introduced two close approaches called the first close approach and the second close approach occur in the same plane but the binary formed and the escaper are not in that plane. Further it is observed that the conjecture of Szebehely (1977) viz. `The measure of escaping orbits is significantly higher than the measure of stable orbits' is likely to be true. The third and last paper offers applications in stellar systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Applications of Two-Parameter Families of Triple Encountersto Stellar Systems (III)

This is the third and last of a series of papers dedicated to the effect of triple encounters with the formation of a binary and escape of the third body on the evolution of stellar systems. Previously obtained results are applied to several astronomical models of triplets formed by sun-like stars, white dwarfs and neutron stars. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of stellar wind,dynamical friction,and star mergers on the dynamical evolution of multiple stars

The dynamical evolution of small stellar groups composed of N=6 components was numerically simulated within the framework of a gravitational N-body problem. The effects of stellar mass loss in the form of stellar wind, dynamical friction against the interstellar medium, and star mergers on the dynamical evolution of the groups were investigated. A comparison with a purely gravitational N-body problem was made. The state distributions at the time of 300 initial system crossing times were analyzed. The parameters of the forming binary and stable triple systems as well as the escaping single and binary stars were studied. The star-merger and dynamical-friction effects are more pronounced in close systems, while the stellar wind effects are more pronounced in wide systems. Star-mergers and stellar wind slow down the dynamical evolution. These factors cause the mean and median semimajor axes of the final binaries as well as the semimajor axes of the internal and external binaries in stable triple systems to increase. Star mergers and dynamical friction in close systems decrease the fraction of binary systems with highly eccentric orbits and the mean component mass ratios for the final binaries and the internal and external binaries in stable triple systems. Star mergers and dynamical friction in close systems increase the fraction of stable triple systems with prograde motions. Dynamical friction in close systems can both increase and decrease the mean velocities of the escaping single stars, depending on the density of the interstellar medium and the mean velocity of the stars in the system.     

Close triple approaches and escape in the three-body problem

We have studied a total of 5000 close triple approaches resulting in escape, for equal-mass systems with zero initial velocities. Escape is shown to take place in the majority of the cases after a fly-by close triple approach when the escaper passes near the centre of mass along an almost straight-line orbit. A number of configurational and kinematical parameters are introduced in order to characterize the triple approach. The distributions of these parameters are investigated. A comparison with 831 examples in the vicinity of the so-called Pythagorean problem is carried out. We find that the general features of close triple approaches which result in escape are the same for both types of systems.     

Approach-ejection correlation in the general three-body problem

The relations between parameters of triple approaches and the lengths of subsequent ejections are analyzed for the general three-body problem with components of equal masses and zero initial velocities. A statistically significant correlation is shown to exist between the closeness of approaches and the lengths of subsequent ejections: closer approaches generally result in longer ejections. We have found several systems that evolve to a temporary quasi-stable chain-like configuration.     

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.     

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.     

Global chaoticity in the Pythagorean three-body problem

The effects of small changes in the initial conditions of the Pythagorean three-body problem are investigated by computer simulations. This problem consists of three interacting bodies with masses 3, 4 and 5 placed with zero velocities at the apices of a triangle with sides 3, 4 and 5. The final outcome of this motion is that two bodies form a binary and the third body escapes. We attempt to establish regions of the initial positions which give regular and chaotic motions. The vicinity of a small neighbourhood around the standard initial position of each body defines a regular region. Other regular regions also exist. Inside these regions the parameters of the triple systems describing the final outcome change continuously with the initial positions. Outside the regular regions the variations of the parameters are abrupt when the initial conditions change smoothly. Escape takes place after a close triple approach which is very sensitive to the initial conditions. Time-reversed solutions are employed to ensure reliable numerical results and distinguish between predictable and non-predictable motions. Close triple approaches often result in non-predictability, even when using regularization; this introduces fundamental difficulties in establishing chaotic regions.     

Strong triple interactions in the general three-body problem

Strong three-body interactions play a decisive role in the course of the dynamical evolution of triple systems having positive as well as negative total energies. These interactions may produce qualitative changes in the relative motions of the components. In triple systems with positive or zero total energies the processes of formation, disruption or exchange of the components of binaries take place as the result of close approaches of the three single bodies or as the result of the passages of single bodies past wide or hard binaries. In the triple systems with negative energies, the strong triple interactions may result in an escape from the system as well as a formation of a hard final binary. This paper summarizes the general results of the studies of the strong three-body interactions in the triple systems with positive and negative energies. These studies were conducted at the Leningrad University Observatory by computer simulations during 1968–1989.     

Binary and triple collisions causing instability in the free-fall three-body problem

Dominant factors for escape after the first triple-encounter are searched for in the three-body problem with zero initial velocities and equal masses. By a global numerical survey on the whole initial-value space, it is found that not only a triple-collision orbit but also a particular family of binary-collision orbits exist in the set of escape orbits. This observation is justified from various viewpoints. Binary-collision orbits experiencing close triple-encounter turn out to be close to isosceles orbits after the encounter and hence lead to escape. Except for a few cases, binary-collision orbits of near-isosceles slingshot also escape. This revised version was published online in July 2006 with corrections to the Cover Date.     

Orbital evolution of the Gefion and Adeona asteroid families: close encounters with massive asteroids and the Yarkovsky effect

Asteroid families are groupings of minor planets identified by clustering in their proper orbital elements; these objects have spectral signatures consistent with an origin in the break-up of a common parent body. From the current values of proper semimajor axes a of family members one might hope to estimate the ejection velocities with which the fragments left the putative break-up event (assuming that the pieces were ejected isotropically). However, the ejection velocities so inferred are consistently higher than N-body and hydro-code simulations, as well as laboratory experiments, suggest. To explain this discrepancy between today’s orbital distribution of asteroid family members and their supposed launch velocities, we study whether asteroid family members might have been ejected from the collision at low speeds and then slowly drifted to their current positions, via one or more dynamical processes. Studies show that the proper a of asteroid family members can be altered by two mechanisms: (i) close encounters with massive asteroids, and (ii) the Yarkovsky non-gravitational effect. Because the Yarkovsky effect for kilometer-sized bodies decreases with asteroid diameter D, it is unlikely to have appreciably moved large asteroids (say those with D > 15 km) over the typical family age (1-2 Gyr).For this reason, we numerically studied the mobility of family members produced by close encounters with main-belt, non-family asteroids that were thought massive enough to significantly change their orbits over long timescales. Our goal was to learn the degree to which perturbations might modify the proper a values of all family members, including those too large to be influenced by the Yarkovsky effect. Our initial simulations demonstrated immediately that very few asteroids were massive enough to significantly alter relative orbits among family members. Thus, to maximize gravitational perturbations in our 500-Myr integrations, we investigated the effect of close encounters on two families, Gefion and Adeona, that have high encounter probabilities with 1 Ceres, by far the largest asteroid in the main belt. Our results show that members of these families spreads in a of less than 5% since their formation. Thus gravitational interactions cannot account for the large inferred escape velocities.The effect of close encounters with massive asteroids is, however, not entirely negligible. For about 10% of the simulated bodies, close encounters increased the “inferred” ejection velocities from sub-100 m/s to values greater than 100 m/s, beyond what hydro-code and N-body simulations suggest are the maximum possible initial ejection velocity for members of Adeona and Gefion with D > 15 km. Thus this mechanism of mobility may be responsible for the unusually high inferred ejection speeds of a few of the largest members of these two families.To understand the orbital evolution of the entire family, including smaller members, we also performed simulations to account for the drift of smaller asteroids caused by the Yarkovsky effect. Our two sets of simulations suggest that the two families we investigated are relatively young compared to larger families like Koronis and Themis, which have estimated ages of about 2 Byr. The Adeona and Gefion families seems to be no more than 600 and 850 Myr old, respectively.     

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.     

On the rotation periods of the components of the triple system TYC 9300-0891-1AB/TYC 9300-0529-1 in the Octans Association

Stellar rotation depends on different parameters such as age, mass, initial chemical composition, initial angular momentum, and environment characteristics. The range of values of these parameters causes the dispersion in the rotation period distributions observed in young stellar clusters/associations. We focus our investigation on the effects of different circumstellar environments on stellar rotation. More specifically, we consider the effects of a perturber stellar companion on the accretion-disc lifetime at early evolution stages.We are searching in stellar Associations for visual triple systems where all stellar parameters are similar, with the only exceptions of the unknown initial rotation period, and of the circum-stellar environment, in the sense that one of the two about equal-mass components has a close-by third ‘perturber’ component.In the present study we analyze the 35-Myr old visual triple system TYC 9300-0891-1AB + TYC 9300-0529-1 in the young Octans stellar association consisting of three equal-mass K0V components. We collected from the literature all information that allowed us to infer that the three components are actually physically bound forming a triple system and are members of the Octans Association. We collected broad-band photometric timeseries in two observation seasons. We discovered that all the components are variable, magnetically active, and from periodogram analysis we found the unresolved components TYC 9300-0891-1AB to have a rotation period P = 1.383 d and TYC9300-0529-1 a rotation period P = 1.634 d.TYC 9300-0891-1A, TYC 9300-0891-1B, and TYC 9300-0529-1 have same masses, ages, and initial chemical compositions. The relatively small 16% rotation period difference measured by us indicates that all components had similar initial rotation periods and disc lifetimes, and the separation of 157 AU between the component A and the ‘perturber’ component B (or vice-versa) has been sufficiently large to prevent any significant perturbation/shortening of the accretion-disc lifetime.     

The present status of periodic orbits

Recent results on periodic orbits are presented and it is shown that the periodic orbits can be used in the study of planetary systems and triple or multiple stellar systems. Triple stellar systems are stable even for close approaches of the three components. Also stable triple systems exist with nearly zero angular momentum. For the planetary systems a global view is obtained from which it is clear which configurations are stable or unstable and also what factors affect the stability. Also, the relation between resonance and instability is studied by making use of periodic orbits.     

Dense star clusters as the sources of gamma-ray-burst progenitors

Two independent sets of arguments lead us to conclude that the progenitors of superintense bursts (with an energy yield larger than that for ordinary supernovae by one or two orders of magnitude) are born in massive dense star clusters, but generally flare up only after they have left the cluster; these are the same objects that are the progenitors of gamma-ray bursts (GRBs). Each of the giant stellar arcs which are grouped into multiple systems of stellar complexes in the LMC and NGC 6946 could only be produced by a single powerful energy release near its center. The progenitors of these systems of arc-shaped stellar complexes must have had a common source nearby, and it could only be a massive star cluster. Such clusters are actually known near both systems. On the other hand, calculations of the dynamical evolution of star clusters show that close binary systems of compact objects are formed in the dense central parts of the clusters and are then ejected from them during triple encounters. Mergers of the components of such systems are believed to be responsible for GRBs. Since their progenitors are ejected from the cluster before merging, the arc-shaped stellar complexes produced by GRBs are observed near (but not around) the parent clusters. If a considerable fraction of the GRB progenitors are formed as a result star encounters in massive star clusters, and if the GRBs themselves trigger star formation near the parent clusters, then observations of GRBs in star-forming regions are consistent with their origin during mergers of pairs of compact objects.     

A model for the chemical evolution of the galaxy with refuses

A model is presented for the chemical evolution of the solar neighbourhood which takes into account three families of galactic objects, according to their condensation states: stars, refuses and gas. Stars are defined as all condensed objects with masses greater than or equal to the minimum mass which ignites hydrogen and which will give rise to an evolutionary track on the HR diagram to the left of Hayashi's limit; refuses include the remnants, which are compact objects resulting from stellar deaths, and the residues, which have masses not large enough to ignite hydrogen; gas is defined as the mass which can be condensed to form stars and/or residues. We have developed equations for the mass evolution of each family, and have studied the gas metallicity distribution within the framework of the instantaneous recycling approximation, adopting different initial conditions. In order to constrain the model parameters we have also used preliminary evaluations of comet cloud masses to investigate the role of the residues as sinks of heavy elements in the Galaxy.     

High-velocity stars from decay of small stellar systems

In this paper we present numerical results on the decay of small stellar systems under different initial conditions (multiplicity 3 ≤  N  ≤ 10, and various mass spectra, initial velocities and initial configurations). The numerical treatment uses the CHAIN1 code (Mikkola &38; Aarseth). Particular attention is paid to the distribution of high-velocity escapers: we define these as stars with velocity above 30 km s⁻¹. These numerical experiments confirm that small N -body systems are dynamically unstable and produce cascades of escapers in the process of their decay. It is shown that the fraction of stars that escape from small dense stellar systems with an escape velocity greater than 30 km s⁻¹ is ∼1 per cent for all systems treated here. This relatively small fraction must be considered in relation to the rate of star formation in the Galaxy in small groups: this could explain some moderately high-velocity stars observed in the Galactic disc and possibly some young stars with relatively high metallicity in the thick disc.     

Third Body Perturbations of Double Stars

We report on the different results concerning the stability of the hierarchical triple systems where a close binary is accompanied by a third star. There are different possible approaches to answer the question of the stability limits for such triple stars: the most direct investigations can be undertaken in integrating numerically the respective equations of motion for many different initial conditions. It is then difficult to take into account all the important parameters like eccentricities, inclination, phases and masses. Analytical approaches and qualitative methods are more approriate to deal with this problem; the respective results confirm the numerically found results that: 1. for prograde orbits the ratio semimajor axis of the inner orbits to the periastron position of the outer orbit is approximately 3.2 2. for retrograde orbits this ratio is just some 10 percents smaller 3. the results are not sensitive in what concerns the masses involved 4. There is a tendency that the inclinations and eccentricities change slightly the stability limits mentioned above. This revised version was published online in August 2006 with corrections to the Cover Date.     

The close approaches and coalescence in triple systems of gravitating masses,I

By computer simulations, the dynamical evolution of plane triple systems of gaseous protogalaxies and galaxies with zero initial velocities has been studied. Inside the regionD of initial configurations some subregions have been revealed corresponding to a coalescence of protogalaxies on the first double approach. The average spin momenta of mergers are approximately equal to those typical of disk galaxies. In triple galaxies, a coalescence on the first double approach does not occur. The presence of significant hidden mass makes the approaches wider and prevents the coalescence of bodies in the systems without a central object. A central pair in a group of galaxies aids to coalescence. Also the change during time of the virial coefficient has been investigated.