Joanna P. Anosova

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.

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.

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.

Computer simulations in the general three-body problem. The theoretical bases of the studies

In this article we present a theoretical method for the study of the general three-body problem by computer simulation developed in the Leningrad State University Astronomical Observatory (LSU AO). This method permits statistical methods to be used for studying the behaviour of triple systems. This is achieved by selecting a representative sample of initial conditions which then reveal general features of the evolution.The main results of numerical experiments on the three-body problem carried out at the LSU AO during the past 25 years have been summarized in the reviews by Anosova (1985), Anosova and Orlov (1985), and Anosova (1986).Systematic studies of about 3 × 104 triple systems with negative total energy (E < 0) have yielded the following main results. Most (93.4%) 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 (4.3%) of 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 equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] = (107.1 ± 1.8) crossing times τ for equal-mass components. Thus, for solar mass components and a typical dimension d = 0.01 pc, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] = (1.6 ± 1.5) × 106 y, and for triple galaxies with M = 101° M0 and d = 50 kpc, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] = (1.8 ± 1.7) × 1011 y. The value % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] decreases with increasing mass dispersion.In this article we also carry out a theoretical analysis of the changes of the integrals of motion in the general three-body problem used as the controls on the calculations. The following basic results have been found: (1) analytical functions of the changes of the integrals of motion during the integration time have been obtained; (2) changes in the integrals of the mass-centre of a triple system do not correlate with the cumulative integration errors; (3) the cumulative changes of the integral of energy are proportional to the sum of squares of the cumulative errors in the coordinates and the velocities of the bodies; (4) the cumulative changes of the square of the total angular momentum are proportional to the product of the square of these cumulative errors.The analysis of the accuracy of computer simulations conducted in LSU AO for the 3 × 104 triple systems with E < 0 is summarized by the following basic qualitative results: (1) the unstable triple systems decay after a mean lifetime % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] ≌ 100τ or % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] ≌ 104 % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGObaaaaaa!3C6A!\[\overline h \]t where τ is a crossing time, and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGObaaaaaa!3C6A!\[\overline h \]τ, is a mean integration step After this integration time % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] the mean cumulative relative changes % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamyraaaaaaa!3D10!\[\overline {DE} \] of the integrals of the energy of the triple systems are equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamyraaaaaaa!3D10!\[\overline {DE} \] = (0.9±0.1) × 10−4, and the mean cumulative relative changes % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamitaaaaaaa!3D17!\[\overline {DL} \] of the area integrals are equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamitaaaaaaa!3D17!\[\overline {DL} \] = (1.0±0.1) × 10−6; the mean values of the cumulative errors % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Gaamiraiaadkhaaaa!3D2C!\[{Dr}\], % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamOvaaaaaaa!3D21!\[\overline {Dv} \] in defining the coordinates (r) and velocities (v) of the bodies (during the total integration time % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \]) are equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamOCaaaaaaa!3D3D!\[\overline {Dr} \] = 0.5 × 10−3d, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamODaaaaaaa!3D41!\[\overline {Dv} \] = 0.5 × 10−2v, where d is the unit of distance, and v is the unit of velocity; the mean local integration errors (of one integration step) are equal to σr= 5 × 10−8d, 6v = 5 × 10−7v; (2) the process of accumulation of integration errors has a complicated character and correlates strongly with the process of dynamical evolution of the triple systems; (a) because of the strong gravitational interplays of the bodies, the process of the accumulation of the integration errors is very intensive; however, the triple s

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.

Distribution of minima of the current size of the triple systems and their disruption

The dynamical evolution of triple systems has been studied by computer simulations. A function ρ(t) has been defined, where p is the maximum distance of the components from their centre of inertia, and t is the time. The value of ρ is used to indicate the current size of the triple system. The minima of ρ have been followed during the course of evolution of the triples. A distribution of f(ρmin) has been obtained, which is described by the following statistical parameters: the mode is equal to 0.65d, the mean value ρmin= 0.750d, r.m.s. is 0.477d, the asymmetry is 0.218, the excess is 2.04 where d is the mean harmonic distance between the bodies in the equilibrium state of the triple system. As a rule, escapes from triples occur only after close three-body approaches.

On actual presence of discordant-redshift galaxies in compact groups

Hicksons compact galaxy groups were classified using the statistical criterion which includes the radial velocities of galaxies as well as their relative positions. These groups on the whole and their components are identified as the confident and probable non-chance ones as well as probable and confident chance ones. All confident chance objects have the discordant radial velocities (the differences of radial velocitiesDV ∈ [1000; 20000] km s−1). The special class of objects ‘bright discordants’ is selected. These galaxies have the discordant radial velocities withDV ∈ [825; 8440] km s−1 and have a strong tendency to be the brightest components of their groups. The lowest difference of radial velocities for the last class of objects 〈DV〉 = (1.0±0.2) × 103 km s−1 and we accept this value of DV as the lowest value of discordant radial velocities. It is found that the biggest part of Hicksons compact groups consist of non-chance aggregations of galaxies and some of the cases of discordant-redshifts require a special study in order to explain their origin from a dynamic or some other point of view.

Search for the matter clumps in scalesZ ~ 1

The diagramV - log(1 +ze) as function of (α, δ) is considered for the quasars. HereV is the apparent visual magnitude,ze is the emission line redshift,α andδ are the equatorial coordinates. Two opposite extreme ‘spots’ NE and SE are observed on the sky, where the inclination of the straight line fitting the dependenceV - log(1 +ze) is maximum and minimum. The coordinates of the centres of these extreme spots are (αNE,δNE) = (282°, +42°) and (αSE,δSE) = (70°, -38°) with errors 5°. A hypothesis of the Superattractor (SA) is proposed to explain such an effect. Two independent tests of this hypothesis are realized. First, the dependence or the frequency a of the absorbers in QSO spectra on (α, δ) is investigated. A region of the larger a is found. The coordinates of its centre are (α, δ) = (82°, - 10°) with error 5°. Second, the cases ofza >ze are plotted in the Mercatorial projection (α, δ). The most of the caseszα -ze > 0.02 are concentrated within the circle with radiusR = 34° and centre (α, δ) = (50°, - 15°). The both anomalous regions overlap the Southern extreme spot around SE. The SA direction is (α, δ) = (67°, -21°) with errors about 12°. The redshift of SA iszSA = 1.7 ± 0.3 that corresponds to the distancerSA = (3100 ± 300)h−1 Mpc for the Hubble constantH0 = 75h kms−1 Mpc−1. The SA mass isMSA ~ 1018-1020M⊙. The orientation of the normal to the quasiperiodical large-scale sheet structure on the sky occurs near SA.

A general scheme of Newtonian interactions in systems of gravitating masses

This Letter reviews the results by computer simulations on the three-body problem carried out at Leningrad University Astronomical Observatory (Anosova, 1986, 1988, 1989). The intensive systematic studies of triple systems with negative and positive total energies have yielded the general features of the evolution of these systems. The processes of formation of the wide and hard binaries have been studied in details. The scenario of the general class of the final motions of the triple systems with negative total energy is considered, the necessary conditions of disruption of these systems are formulated.

Close Approaches and Coalescence in the Triple Systems of Gravitating Masses

The dynamical evolution of the plane triple systems of the gaseous protogalaxies and galaxies with zero initial velocities has been studied by computer simulations. Inside the region D of initial configurations there have been revealed some subregions corresponding to a coalescence of protogalaxies on the first double encounter of them. The average spin momenta of “mergers” are approximately equal to the typical ones of the disk galaxies observed. In the triple galaxies, a coalescence on the first double approach does not occur. A presence of a big hidden mass makes the encounters wider and prevents some coalescences of bodies in the systems without a central object. The central pair in a group of galaxies aids to the coalescences. A conduct of virial ratio was studied.