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.     

Dynamical evolution of triple systems with big differences in the masses of bodies. A criterion for stability

The dynamical evolution of about 1.5 million planar hierarchical triple systems with a negative total energy and different-mass bodies is investigated by computer simulations. We considered both cases — prograde and retrograde motions of bodies. For every system, calculations were carried out either till a time when the Marchal'set al. (1984) criterion of escape of a body from a triple system was satisfied (the unstable triple systems) or during 1000 rotations of a total system (the stable triple systems). Computations were carried out on three computers-Sunstations in the Physical Research Laboratory, Ahmedabad, India during several months continuously. We changed smoothly the initial value of the coefficient of hierarchy of triples $$q = r_{3 - 12} /r_{12} $$ Wherer ₁₂ is a distance between close bodiesM1,M2 andr _3–12 is a distance between their center of masses and a distant bodyM3. We define critical (minimum) values of the coefficientq of hierarchy of stable triple systems with a relative accuracy δq=1%. Ratios of masses of bodies belong to the interval [0.13, 244.00]. A possibility of extention of these results for hierarchical subsystems with different multiplicities inside clusters is discussed.     

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.     

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.     

Dynamical stability of triple stars

The dynamical stability of 38 observed hierarchical triple stars with known orbital elements of the internal and external binary subsystems and component masses is considered. Four different criteria of dynamical stability are used. The observed stability parameters and their critical values are calculated by taking into account errors in the orbital elements and component masses. Most triple systems are stable. According to some criteria, several triple stars (ADS 440, ξ Tau, λ Tau, ADS 3358, VV Ori, ADS 10157, HZ Her, Gliese 795, ADS 15971, and ADS 16138) may be dynamically unstable. This result is probably associated with unreliability of the empirical stability criteria and/or with errors in the observed quantities.     

Chaos in N-body systems

Here is a selection of applications of what is now called theory of dynamical systems in galactic dynamics and N-body systems. The study of chaotic motions in potentials used as a model for elliptical galaxies is a first example of these applications. The interest in this problem stems from the fact that there are now many theoretical and observational evidences that the overall potentials of galaxies are indeed non-integrable. There are classes of objects, for example small and intermediate luminosity elliptical galaxies, for which the presence of the famous third integral is not necessary or others in which we observe peculiarities in their photometry or kinematics. We address here some of these issues and their implications in modifying our current understanding of the structure and evolution of galaxies.More in general, there is the natural question of how the systems we see have settled to their present status and what would happen if some external cause perturbs it. This issue is related to the question of the stochasticity involved in the general N-body dynamics, especially when N is very large. An N-body dynamical system is definitely chaotic, as shown by several numerical investigations, at least for N not very large. However, this statement must be reconciled with the picture of non-collisional equilibrium of big systems. The second part of this review presents a survey of numerical experiments and an interpretation of the results obtained using standard chaoticity indicators.     

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.     

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.     

Dynamics of double open clusters in the galactic field

Data from catalogs of multiple open star clusters (OSCs) have been used to compile a list of double and triple OSCs. Seven pairs of young OSCs with similar ages of their components have been selected from them. The dynamical evolution of the selected pairs of clusters in the Galactic gravitational field has been simulated numerically. The individual cluster masses have been estimated and the time dependences of the separations between the clusters have been constructed. The separations between the clusters are shown to exceed the tidal cluster radii. Various hypotheses of the origin of double OSCs are discussed: chance formations of pairs, formation within the same star complex, etc.     

Searching for the source of short-period comet nuclei: Choosing between different asteroid groups

This study continues our previous works on searching for the main source of the nuclei of Jupiter family comets (JFCs). Angular orbit element distributions are analyzed for comets and asteroids of different groups. The distributions of JFCs by argument of perihelion ω and longitude of perihelion π are studied. The distributions are shown not to have been formed during the evolution of JFCs in their current orbits. Similar distributions N(ω) and N(π) are not observed in bodies that have come into the JFC orbits from external sources. At the same time, the distributions of JFCs by all angular orbit elements are very similar to those of the Trojans. It is concluded that the latter are likely to be the main source of the JFC nuclei.     

An implementation ofN-body chain regularization

The chain regularization method (Mikkola and Aarseth 1990) for high accuracy computation of particle motions in smallN-body systems has been reformulated. We discuss the transformation formulae, equations of motion and selection of a chain of interparticle vectors such that the critical interactions requiring regularization are included in the chain. The Kustaanheimo-Stiefel (KS) coordinate transformation and a time transformation is used to regularize the dominant terms of the equations of motion. The method has been implemented for an arbitrary number of bodies, with the option of external perturbations. This formulation has been succesfully tested in a generalN-body program for strongly interacting subsystems. An easy to use computer program, written inFortran, is available on request.     

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 (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 is used to indicate the current size of the triple system. The minima of have been followed during the course of evolution of the triples. A distribution of f(_min) has been obtained, which is described by the following statistical parameters: the mode is equal to 0.65d, the mean value _min= 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.     

Periodic orbits in the general three-body problem and the relationship between them

We study the regions of finite motions in the vicinity of three simple stable periodic orbits in the general problem of three equal-mass bodies with a zero angular momentum. Their distinctive feature is that one of the moving bodies periodically passes through the center of mass of the triple system. We consider the dynamical evolution of plane nonrotating triple systems for which the initial conditions are specified in such a way that one of the bodies is located at the center of mass of the triple system. The initial conditions can then be specified by three parameters: the virial coefficient k and the two angles, φ₁ and φ₂, that characterize the orientation of the velocity vectors for the bodies. We scanned the region of variation in these parameters k∈(0, 1); φ₁, φ₂∈(0, π) at steps of δk=0.01; δφ₁=δφ₂=1° and identified the regions of finite motions surrounding the periodic orbits. These regions are isolated from one another in the space of parameters (k, φ₁, φ₂). There are bridges that correspond to unstable orbits with long lifetimes between the regions. During the evolution of these metastable systems, the phase trajectory can “stick” to the vicinity of one of the periodic orbits or move from one vicinity to another. The evolution of metastable systems ends with their breakup.     

Dynamical evolution of two associated galactic bars

We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple Periodic Orbits in the Ring Problem: Families of Triple Periodic Orbits

The ring problem deals with the motion of a small body which is subjected to the combined gravitational attraction of N massive bodies arranged in an annular configuration. In this paper we study the distribution of the triple periodic orbits in the phase space of the initial conditions and we discuss their evolution and their principal features. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of Planetary Migration on Natural Satellites of the Outer Planets

Numerous studies in the past few years have analyzed possible effects of planetary migration on the small bodies of the Solar System (mainly asteroids and KBOs), with the double aim of explaining certain dynamical structures in these systems, as well as placing limits on the magnitude of the radial migration of the planets. Here we undertake a similar aim, only this time concentrating on the dynamical stability of planetary satellites in a migration scenario. However, different from previous works, the strongest perturbations on satellite systems are not due to the secular variation of the semimajor axes of the planets, but from the planetesimals themselves. These perturbations result from close approaches between the planetesimals and satellites.We present results of several numerical simulations of the dynamical evolution of real and fictitious satellite systems around the outer planets, under the effects of multiple passages of a population of planetesimals representing the large-body component of a residual rocky disk. Assuming that this component dominated the total mass of the disk, our results show that the present systems of satellites of Uranus and Neptune do not seem to be compatible with a planetary migration larger than even one quarter that suggested by previous studies, unless these bodies were originated during the late stage of evaporation of the planetesimal disk. For larger variations of the semimajor axes of the planets, most of the satellites would either be ejected from the system or suffer mutual collisions due to excitation in their eccentricities. For the systems of Jupiter and Saturn, these perturbations are not so severe, and even large migrations do not introduce large instabilities.Nevertheless, even a small number of 1000-km planetesimals in the region may introduce significant excitation in the eccentricities and inclinations of satellites. Adequate values of this component may help explain the present dynamical distribution of distant satellites, including the highly peculiar orbit of Nereid.     

Evolution of Planetesimal Velocities Based on Three-Body Orbital Integrations and Growth of Protoplanets

We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10-30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies.     

The Influence of a Mass Spectrum on the Long Term Evolution of a Self-Gravitating System of Softened Particles

Using N-body simulations, we study the effects of the mass spectrum in the evolution of self-gravitating systems of softened point-mass particles. The mass function is described by a power law and the ratio between the maximum and minimum mass is . We showed that the dynamical evolution of the system depends on the mass spectrum: the secular evolution time is longer for flatter mass spectrum. For the steepest mass spectrum, the secular evolution time is of the order of the relaxation time. The mass segregation effects are achieved rapidly and the core-halo structures are formed. The projected number distributions for the systems with mass spectrum change drastically with the evolution while the projected mass distributions are not affected. Velocity dispersion profiles are modified in the sense of heating of the central regions of the systems, while the velocity anisotropy profiles are slightly affected. The consequence of our results on the dynamical evolution of clusters of galaxies is presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamical states and dynamical evolution of the selected triple star systems

The general problem of investigating multiple stellar systems is formulated. It is shown that the complete solution of this problem requires 1) a complex of astrometric and astrophysical observations of multiple stars, and the 2) maximum attainable precision of such observations. The conditions under which this precision can be achieved are discussed.The most important characteristics of the dynamical states of multiple systems are the total energy E and the relative energies E_ij of the bodies in these systems. For eight triple systems (ADS 1630, 2926, 6175, 6650, 6811, 7114, 9626, 9909), statistical tests — a method of calculating the uncertainties _E and _Eij from the errors of the observational data — are used to find the probabilities for dynamical states and the values E_tr±E_tr and E_bin±E_bin.Only three triple stars appear certain to be physical systems with a dynamical connection between the components — ADS 6175 and 9909 with the probabilities P>0.80 are dynamical non-hierarchical unstable triple systems with a complicated motions of the components; the final state of these systems is an escape. In the triple system ADS 1630 a qualitative course of the component motions has not been determined because of the larger errors in the observed data. The dynamical evolution of the triple system ADS 9909 is under study.