Star cluster evolution with primordial binaries. I - A comparative study
Abstract
The evolution of equal-mass star clusters containing a mass fraction of
about 20 percent binaries has been followed using direct integration,
making one run each for a total number of stars of N = 282 and N = 563,
and four runs for N = 1126. For comparison the evolution of an
equivalent star system where the binaries were replaced by stars twice
as heavy as the other stars was followed. The pre-core-collapse
evolution is driven by mass segregation between the equal-mass single
stars and the binaries, which are twice as heavy. After core collapse,
the cluster shows, on average, a smooth reexpansion driven by a steady
rate of burning (hardening) of primordial binaries. With so much
primordial fuel present, the postcollapse cluster core is significantly
larger than is the case in comparison runs without primordial binaries.
- We study analytically and experimentally certain symplectic and time-reversible N-body integrators which employ a Kepler solver for each pair-wise interaction, including the method of Hernandez & Bertschinger (2015). Owing to the Kepler solver, these methods treat close two-body interactions correctly, while close three-body encounters contribute to the truncation error at second order and above. The second-order errors can be corrected to obtain a fourth-order scheme with little computational overhead. We generalise this map to an integrator which employs a Kepler solver only for selected interactions and yet retains fourth-order accuracy without backward steps. In this case, however, two-body encounters not treated via a Kepler solver contribute to the truncation error.
- We investigate effects of hardness of primordial binaries on whole evolution of star clusters by means of N-body simulations. Using newly developed code, GORILLA, we simulated eleven N=16384 clusters with primordial binaries whose binding energies are equal in each cluster in range of 1-300kT_0, where 1.5kT_0 is average stellar kinetic energy at the initial time. We found that, in both soft (< 3kT_0) and hard (> 300kT_0) limits, clusters experience deep core collapse. In the intermediate hardness (10-100kT_0), the core collapses halt halfway due to an energy releases of the primordial binaries. The core radii at the halt can be explained by their energy
- The evolution of globular cluster models containing significant numbers of primordial binaries is followed to the point at which the primordial fuel is spent and the behavior reverts to the 'single-star' form. This point is reached after some 30 initial core collapse times when the cluster has expanded by a factor of about five to seven. For N about 1100, the ratio of core radius to virial radius just after core collapse is about 0.12. This ratio then drops linearly in comoving time, reaching the asymptotic single-star value of about 0.05. The core mass drops by a factor of two during this time. The first detailed statistical study of the spatial evolution of the binary population in a star cluster is also reported. The mass fraction of binaries in the core shows a steady, near-linear decrease from 0.5 to 0.1 between 20 and 250 instantaneous half-mass crossing times. Hard binaries are found to harden both through the effects of strong scattering and through the declining energy scale set by the expanding cluster.
- We investigated whether gravothermal expansion occurs in N-body systems. As the most idealized case, we studied the evolution of an N-body system in a spherical adiabatic wall which is initially in thermal equilibrium. Such a system is thermodynamically unstable if the central density is more than 709 times higher than the density just inside the wall. We performed simulation of a 2048-body system with and without artificial perturbations. We found that gravothermal expansion took place as well as gravothermal contraction. Whether gravothermal expansion occurs or not is determined by the perturbation or statistical fluctuation on the temperature structure in the initial model. The results were compared with that of a self-gravitating gas system.
- Results are presented of comparisons between star-count mass functions and surface density profiles for the globular cluster M71 and multimass, tidally truncated Fokker-Planck simulations reheated by three-body binaries. The degree of mass segregation and the short relaxation time observed for the cluster suggest that M71 should be a postcore-collapse cluster. It is shown that gravothermal oscillations are unlikely to affect the comparison between observations and theory except in the case of clusters with extreme cusps. The presence of massive stellar remnants can flatten the postcore-collapse surface brightness profile, but such models fail to reproduce the observed mass segregation and also predict an unacceptably high value of the central velocity dispersion. Models in which the heating rate is artificially enhanced are able to reproduce the observations, but, in the absence of an identified source for this extra heating, such models are not physically justified. It is argued that this type of Fokker-Planck model, in which postcore-collapse expansion is driven solely by three-body binaries, is incomplete and that additional physics, such as the effects of stellar evolution or primordial binaries, is required.
- We review the tidal capture process and in particular the chaotic orbital evolution which follows capture. We discuss the formation of low-mass X-ray binaries in globular clusters via tidal capture and speculate on the possibility that some field low-mass X-ray binaries were formed this way in open clusters which have since dispersed, or in existing old open clusters which are not accessible to observation because of obscuration by dust or because they are indistinguishable from the rich background of galactic stars.
- We report results of a large number of collisional N- body simulations including a mass spectrum and a tidal field. We emphasize the ways in which evolution differs from that observed in isolated systems, by comparison with earlier papers in this series. We show that the evolution hardly alters if the tidal field is replaced by a tidal cut-off. Core collapse is almost unaffected by the tide, but the subsequent post- collapse expansion is eventually reversed by the time that tidal limitation becomes important. As in isolated models, mass segregation almost stops by the end of core collapse; after core bounce there is a substantial increase in the mean mass, caused by preferential escape of stars of low mass. The early phase of rapid mass segregation is also characterized by an approach to equipartition in the inner parts of the system, although the outer parts remain far from equipartition. Even here, however, there is a slow tendency to equipartition in post-collapse expansion, as the tidal stripping of the outer parts gradually exposes parts of the cluster where equipartition was partially established. Similar remarks apply to the evolution of the anisotropy. The evolution of the core is virtually unaffected by the tide until well after core bounce; in late post-collapse evolution there is a tendency for the mass of the core to decrease slightly, whereas it is nearly constant in isolated models. Similarly, in the presence of a tide the mean number of binaries in the system tends to decrease after the end of core collapse. The total internal binding energy in bound binaries is smaller than at comparable stages of post-collapse evolution in isolated systems. In the overall energy budget, one obvious difference between isolated and tidally limited models is that the energy of escaping single stars is initially negative, although it becomes positive later as the energy carried off by ejecta from three-body interactions becomes dominant.
- Coravel observations of 100 F5-K0 stars in the Pleiades cluster yielded 13 spectroscopic binary stars, and 11 orbits were determined. All 11 periods are shorter than 1000 days and the longest circular period is 7.05 days. One single-lined spectroscopic binary belongs to a triple system, the orbital motion of which has been detected. Based on a complete sample in the color range B-V = 0.40 - 0.90, (88 stars), the percentage of spectroscopic binaries with P less than 1000 days is 13 percent. The number of single:binary:triple stars is 56:30:2.
- The importance of primordial binaries for the dynamical evolution of isolated star clusters is studied, emphasizing developments after core collapse. Fokker-Planck methods are used to represent the distribution of binary and single-star translational energies accurately and with the minimum of numerical noise while the central density of the cluster varies over several orders of magnitude. Mass stratification and modest core contraction occur first, followed by a period during which the core radius is nearly constant, and then by gravothermal oscillations. Even a large initial binary population cannot prevent initial contraction of the core radius with respect to the half-mass radius if the initial model has low concentration. On the other hand primordial binaries can support the core against further collapse for many half-mass relaxation times.
- An extensive series of three-body interactions involving hard binaries and single stars are calculated by direct integration of ˜105 encounters. Unlike previous calculations, the stars have different mass ratios, chosen to be representative of the stellar and remnant population in a globular cluster, and most effort was devoted to the difficult case of hard binaries, the most likely primordial binaries to survive in a cluster. Graphs, tables and analytical fits to differential and integral cross sections for properties of different interaction channels are presented and discussed. The properties include exchange, binary semi-major axis, eccentricity, and vector angular momentum, binary recoil velocity, and dissipative encounters (collision or tidal capture). We compute mean energy transfer as a function of the field star velocity and binary properties (relevant for Fokker-Planck models of the evolution of clusters containing binaries). We find that mass-ratios of order 2 lead to significant differences and new phenomena in the interactions compared with the equal mass case. Resonances are followed and are found to contribute strongly to the cross sections for physical collisions of stars. We find that for moderately hard binaries, exchange of heavy field stars is the dominant process. After such an exchange the physical cross section for subsequent interactions increases, even though the binary is hardened (binding energy increased) by the interaction. For very hard binaries, dissipative interactions dominate the cross section.
- A number of globular clusters appear to have undergone core collapse, in the sense that their predicted collapse times are much shorter than their current ages. Simulations with gas models and the Fokker-Planck approximation have shown that the central density of a globular cluster after the collapse undergoes nonlinear oscillation with a large amplitude (gravothermal oscillation). However, the question whether such an oscillation actually takes place-in real N-body systems has remained unsolved because an N-body simulation with a sufficiently high resolution would have required computing resources of the order of several GFLOPS-yr. In the present paper, we report the results of such a simulation performed on a dedicated special-purpose computer, GRAPE-4. We have simulated the evolution of isolated point-mass systems with up to 32, 768 particles. The largest number of particles reported previously is 10, 000. We confirm that gravothermal oscillation takes place in an N-body system. The expansion phase shows all the signatures that are considered to be evidence of the gravothermal nature of the oscillation. At the maximum expansion, the core radius is ∼1% of the half-mass radius for the run with 32, 768 particles. The maximum core size, rc, depends on N as 〈r〉. ∝ N-1/3. © 1996. The American Astronomical Society. All rights reserved.
- The dynamical evolution of dense clusters of compact stars is studied using direct N-body simulations. The formation of binaries and their subsequent merging by gravitational radiation emission is important to the evolution of such clusters. Aarseth's NBODY5 N-body simulation code is modified to include the lowest order gravitational radiation force during two-body encounters and to handle the decay and merger of radiating binaries. It is used to study the evolution of small-N (= 1000) clusters with different initial velocity dispersions. The initial evolution is similar to that obtained by Quinlan & Shapiro (1989) using a multimass Fokker-Planck code and shows orderly formation of heavy objects. However, the late evolution differs qualitatively from previous results. In particular, we find runaway growth for the most massive object in the cluster: it acquires a mass much larger than that of the other objects and is detached from the smooth mass spectrum of the rest of the objects. We discuss why the Fokker-Planck equation with a mean-rate approach to the merger process cannot model runaway growth, and we present arguments to show that merger by gravitational radiation is expected to be unstable to runaway growth. The results suggest that a seed massive black hole can be formed by runaway growth in a dense cluster of compact stars. The possibility of runaway growth in dense clusters of normal stars is also discussed.
- In this paper, we describe the architecture and performance of the GRAPE-4 system, a massively parallel special-purpose computer for N-body simulation of gravitational collisional systems. The calculation cost of N-body simulation of collisional self-gravitating system is O(N3). Thus, even with present-day supercomputers, the number of particles one can handle is still around 10,000. In N-body simulations, almost all computing time is spent calculating the force between particles, since the number of interactions is proportional to the square of the number of particles. Computational cost of the rest of the simulation, such as the time integration and the reduction of the result, is generally proportional to the number of particles. The calculation of the force between particles can be greatly accelerated by means of a dedicated special-purpose hardware. We have developed a series of hardware systems, the GRAPE (GRAvity PipE) systems, which perform the force calculation. They are used with a general-purpose host computer which performs the rest of the calculation. The GRAPE-4 system is our newest hardware, completed in 1995 summer. Its peak speed is 1.08 TFLOPS. This speed is achieved by running 1692 pipeline large-scale integrated circuits (LSIs), each providing 640 MFLOPS, in parallel.
- We present a new parallel supercomputer implementation of the Monte Carlo method for simulating the dynamical evolution of globular star clusters. Our method is based on a modified version of Hénon's Monte Carlo algorithm for solving the Fokker-Planck equation. Our code allows us to follow the evolution of a cluster containing up to 5 × 105 stars to core collapse in 40 hours of computing time. In this paper we present the results of test calculations for clusters with equal-mass stars, starting from both Plummer and King model initial conditions. We consider isolated as well as tidally truncated clusters. Our results are compared to those obtained from approximate, self-similar analytic solutions, from direct numerical integrations of the Fokker-Planck equation, and from direct N-body integrations performed on a GRAPE-4 special-purpose computer with N = 16384. In all cases we find excellent agreement with other methods, establishing our new code as a robust tool for the numerical study of globular cluster dynamics using a realistic number of stars.
- We present N-body models to complement deep imaging of the metal-poor core-collapsed cluster NGC 6397 obtained with the Hubble Space Telescope. All simulations include stellar and binary evolution in step with the stellar dynamics and account for the tidal field of the Galaxy. We focus on the results of a simulation that began with 100, 000 objects (stars and binaries), 5% primordial binaries, and Population II metallicity. After 16 Gyr of evolution, the model cluster has about 20% of the stars remaining and has reached core collapse. We compare the color-magnitude diagrams of the model at this age for the central region and an outer region corresponding to the observed field of NGC 6397 (about 2-3 half-light radii from the cluster center). This demonstrates that the white dwarf (WD) population in the outer region has suffered little modification from dynamical processes—contamination of the luminosity function by binaries and WDs with non-standard evolution histories is minimal and should not significantly affect measurement of the cluster age. We also show that the binary fraction of main-sequence stars observed in the NGC 6397 field can be taken as representative of the primordial binary fraction of the cluster. For the mass function (MF) of the main-sequence stars, we find that although this has been altered significantly by dynamics over the cluster lifetime, especially in the central and outer regions, the position of the observed field is close to optimal for recovering the initial MF of the cluster stars (below the current turn-off mass). More generally we look at how the MF changes with radius in a dynamically evolved stellar cluster and suggest where the best radial position to observe the initial MF is for clusters of any age. We discuss computational constraints that restrict the N-body method to non-direct models of globular clusters currently, how this affects the interpretation of our results regarding NGC 6397, and future plans for models with increased realism.
- We report on the results of a direct N-body simulation of a star cluster that started with N = 200 000, comprising 195 000 single stars and 5000 primordial binaries. The code used for the simulation includes stellar evolution, binary evolution, an external tidal field and the effects of two-body relaxation. The model cluster is evolved to 12 Gyr, losing more than 80 per cent of its stars in the process. It reaches the end of the main core-collapse phase at 10.5 Gyr and experiences core oscillations from that point onwards – direct numerical confirmation of this phenomenon. However, we find that after a further 1 Gyr the core oscillations are halted by the ejection of a massive binary comprised of two black holes from the core, producing a core that shows no signature of the prior core-collapse. We also show that the results of previous studies with N ranging from 500 to 100 000 scale well to this new model with larger N. In particular, the time-scale to core collapse (in units of the relaxation time-scale), mass segregation, velocity dispersion and the energies of the binary population all show similar behaviour at different N.
- At the moment of deepest core collapse, a star cluster core contains less than ten stars. This small number makes the traditional treatment of hard binary formation, assuming a homogeneous background density, suspect. In a previous paper, we have found that indeed the conventional wisdom of binary formation, based on three-body encounters, is incorrect. Here we refine that insight, by further dissecting the subsequent steps leading to hard binary formation. We find that the conventional treatment does remain valid for direct three-body scattering, but fails for resonant three-body scattering. Especially democratic resonance scattering, which forms an important part of the analytical theory of three-body binary formation, takes too much space and time to be approximated as being isolated, in the context of a cluster core around core collapse. We conclude that, while three-body encounters can be analytically approximated as isolated, subsequent strong perturbations typically occur whenever those encounters give rise to democratic resonances. We present analytical estimates postdicting our numerical results. If we only had been a bit more clever, we could have predicted this qualitative behaviour.
- We describe the implementation of tidal circularization of binaries in an N-body code for star cluster simulations. The first part contains the theoretical framework for normal and chaotic tidal interactions, including capture from hyperbolic orbits. This formulation yields convenient expressions which are used to modify the binary elements. Stars are represented as polytropes, with a time-dependent effective polytropic index calculated for evolving stars. Stellar evolution is treated using a fast look-up table for stellar masses and radii. This gives a consistent astrophysical description of open clusters containing a significant proportion of primordial binaries with a wide range of masses and periods. An analytic expression for the chaos boundary for arbitrary mass ratio and polytropic indices is presented. We provide detailed correction procedures for tidal circularization and chaotic motion for perturbed binaries which are studied by the classical Kustaanheimo–Stiefel two-body regularization method and also outline a similar treatment for multiple regularization of temporary subsystems involving 3–6 members. Strong interactions in the latter lead to the formation of chaotic binaries and stable hierarchical systems in which the eccentricity of the inner binary may be subject to systematic changes on relatively short time-scales. Finally, we illustrate the effect of tidal circularization by presenting some results of a realistic cluster simulation involving 104 single stars and 500 primordial binaries.
- Though about 80 pulsar binaries have been detected in globular clusters so far, no pulsar has been found in a triple system in which all three objects are of comparable mass. Here, we present predictions for the abundance of such triple systems, and for the most likely characteristics of these systems. Our predictions are based on an extensive set of more than 500 direct simulations of star clusters with primordial binaries, and a number of additional runs containing primordial triples. Our simulations employ a number Ntot of equal-mass stars from Ntot= 512 to 19 661 and a primordial binary fraction from 0 to 50 per cent. In addition, we validate our results against simulations with N= 19 661 that include a mass spectrum with a turn-off mass at 0.8 M⊙, appropriate to describe the old stellar populations of Galactic globular clusters. Based on our simulations, we expect that typical triple abundances in the core of a dense cluster are two orders of magnitude lower than the binary abundances, which in itself already suggests that we do not have to wait too long for the first comparable-mass triple with a pulsar to be detected.
- [abridged] We extend our analysis of the dynamical evolution of simple star cluster models, in order to provide comparison standards that will aid in interpreting the results of more complex realistic simulations. We augment our previous primordial-binary simulations by introducing a tidal field, and starting with King models of different central concentrations. We present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, and N values from 512 to 16384. We also attempt to extrapolate some of our results to the larger number of particles that are necessary to model globular clusters. We characterize the steady-state `deuterium main sequence' phase in which primordial binaries are depleted in the core in the process of `gravitationally burning'. In this phase we find that the ratio of the core to half-mass radius, r_c/r_h, is similar to that measured for isolated systems. In addition to the generation of energy due to hardening and depletion of the primordial binary population, the overall evolution of the star clusters is driven by a competing process: the tidal disruption of the system. We find that the depletion of primordial binaries before tidal dissolution of the system is possible only if the initial number is below 0.05 N, in the case of a King model with W_0=7 and N=4096 (which is one of our longest living models). We compare our findings, obtained by means of direct N-body simulations but scaled, where possible, to larger N, with similar studies carried out by means of Monte Carlo methods.
- In order to interpret the results of complex realistic star cluster simulations, which rely on many simplifying approximations and assumptions, it is essential to study the behaviour of even more idealized models, which can highlight the essential physical effects and are amenable to more exact methods. With this aim, we present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0–100 per cent, with values of N ranging from 256 to 16 384. This allows us to extrapolate the main features of the evolution to systems comparable in particle number with globular clusters. In this range, we find that the steady-state ‘deuterium main sequence’ is characterized by a ratio of the core radius to half-mass radius that follows qualitatively the analytical estimate by Vesperini & Chernoff, although the N dependence is steeper than expected. Interestingly, for an initial binary fraction f greater than 10 per cent, the binary heating in the core during the post-collapse phase almost saturates (becoming nearly independent of f), and so little variation in the structural properties is observed. Thus, although we observe a significantly lower binary abundance in the core with respect to the Fokker–Planck simulations by Gao et al., this is of little dynamical consequence. At variance with the study of Gao et al., we see no sign of gravothermal oscillations before 150 half-mass relaxation times. At later times, however, oscillations become prominent. We demonstrate the gravothermal nature of these oscillations.
- M4 and NGC 6397 are two very similar galactic globular clusters, which differ mainly in their surface brightness profile. M4 has a classic King-like profile, whereas NGC 6397 has a more concentrated profile, which is often interpreted as that of a post-core collapse cluster. In previous papers, however, the authors found that M4 is also a post-core collapse cluster, and concluded that the main reason for the difference between the two surface brightness profiles is fluctuations. This conclusion was reached on the basis of Monte Carlo models, however, and in the present Letter we verify that similar fluctuations occur in N-body models. The models were initialized by generating initial conditions from the Monte Carlo model of NGC 6397 at the simulated age of 12 Gyr, and one was followed for 1 Gyr. The new models help us to clarify the nature of the fluctuations, which have the nature of semiregular oscillations with a time-scale of the order of 108 years. They are influenced by the dynamical role which is played by primordial binaries in the evolution of the core.
- The evolution is described of an idealized star cluster in which all stars have the same mass and initially a modest fraction of the stars form hard binaries. The treatment ranges from simple theory based on considerations of time-scales and energetics, through continuum models which include the effects of segregation of binaries, to N-body simulations of systems with a few thousand stars. Particular consideration is given to the circumstances at the close of core collapse and its immediate aftermath.
- Recent observational searches suggest that the frequency of primordial binaries in globular clusters may reach ~ 10% (see Hut et al. 1992 for a review). Several different treatments conclude that primordial binaries are effective in halting core collapse, supporting the core and driving the post-core collapse expansion phase (Goodman & Hut 1989, McMillan et al. 1990, 1991, Gao et al. 1991, Heggie & Aarseth 1992, McMillan & Hut 1994). The abundance and binding energy distribution have a direct impact on observable characteristics of globular clusters such as the size of the core radius (Vesperini & Chernoff 1994). In this analysis we have investigated how the initial binary distribution function may be altered in the formation of a cluster; the key question we have addressed is whether it is possible for binaries, assumed to be primordial, to survive the birth of the cluster. A detailed description of our results is in Vesperini & Chernoff (1995).
- The problem of the origin of the blue stragglers and contact binaries in the old open clusters M67 and NGC 188 is examined. The hypothesis that blue stragglers and contact binaries are formed via physical stellar collisions and tidal captures, respectively, during strong dynamical interactions involving binary stars is considered. This mechanism can naturally account for many of the observed properties of these objects. An attempt is made to test the collisional hypothesis by carrying out binary-binary and binary-single scattering experiments tailored to the old open clusters M67 and NGC 188. The experiments indicate that isolated interactions involving binaries composed of main-sequence stars are unlikely to account for 100 percent of the blue stragglers and contact binaries in the two clusters. However, such interactions can easily produce about 10 percent of the observed objects. The ways in which the production rate of blue stragglers and contact binaries via collisions may be enhanced are described.
- This paper concentrates on four key tools for performing star cluster simulations developed during the last decade which are sufficient to handle all the relevant dynamical aspects. First we discuss briefly the Hermite integration scheme which is simple to use and highly efficient for advancing the single particles. The main numerical challenge is in dealing with weakly and strongly perturbed hard binaries. A new treatment of the classical Kustaanheimo-Stiefel two-body regularization has proved to be more accurate for studying binaries than previous algorithms based on divided differences or Hermite integration. This formulation employs a Taylor series expansion combined with the Stumpff functions, still with one force evaluation per step, which gives exact solutions for unperturbed motion and is at least comparable to the polynomial methods for large perturbations. Strong interactions between hard binaries and single stars or other binaries are studied by chain regularization which ensures a non-biased outcome for chaotic motions. A new semi-analytical stability criterion for hierarchical systems has been adopted and the long-term effects on the inner binary are now treated by averaging techniques for cases of interest. These modifications describe consistent changes of the orbital variables due to large Kozai cycles and tidal dissipation. The range of astrophysical processes which can now be considered by N-body simulations include tidal capture, circularization, mass transfer by Roche-lobe overflow as well as physical collisions, where the masses and radii of individual stars are modelled by synthetic stellar evolution.
- The formation and evolution of galaxy bulges is one of the most debated topics in the modern astrophysics. One approach to address this issue is to look at the Galactic bulge since it is the closest. According to some theoretical models, our bulge may have built up from the merger of substructures formed from the fragmentation of a gaseous disk in the early phases of Galactic evolution. We may have discovered the remnant of one of these substructures in the stellar system Terzan 5. In fact, Terzan 5 hosts two stellar populations with quite different iron abundances, thus suggesting it once was far more massive than today. Moreover, its peculiar chemistry strikingly resembles that observed in the Galactic bulge. In this Thesis we performed a detailed photometric and spectroscopic analysis of this cluster to determine its formation and evolution. Form the photometric point of view we built a high-resolution differential reddening map in the direction of the system and we measured relative proper motions to separate its member population from the contaminating bulge and disk stars. From the spectroscopic point of view we measured abundances for more than 600 stars belonging to Terzan 5 and its surroundings in order to build the largest field-decontaminated metallicity distribution for this system. We find that the metallicity distribution is extremely broad (>1 dex) and we discovered a third, metal-poor and alpha-enhanced population with average [Fe/H]=-0.8 dex. The striking similarity between Terzan 5 and the bulge in terms of their chemical formation and evolution that is clearly revealed by this Thesis suggests that Terzan 5 formed in situ with the Galactic bulge. In particular its metal-poor populations trace the early stages of the bulge formation, while its most metal-rich component may contain crucial information on the bulge more recent evolution.
- We use direct N-body calculations to investigate the impact of primordial mass segregation on the size scale and mass-loss rate of star clusters in a galactic tidal field. We run a set of simulations of clusters with varying degrees of primordial mass segregation at various galactocentric radii and show that, in primordially segregated clusters, the early, impulsive mass-loss from stellar evolution of the most massive stars in the innermost regions of the cluster leads to a stronger expansion than for initially non-segregated clusters. Therefore, models in stronger tidal fields dissolve faster due to an enhanced flux of stars over the tidal boundary. Throughout their lifetimes, the segregated clusters are more extended by a factor of about 2, suggesting that (at least) some of the very extended globular clusters in the outer halo of the Milky Way may have been born with primordial mass segregation. We finally derive a relation between star–cluster dissolution time, Tdiss, and galactocentric radius, RG, and show how it depends on the degree of primordial mass segregation.
- The evolution of self-gravitating rotating dense stellar systems (e.g. globular clusters, galactic nuclei) with embedded black holes is investigated. The interaction between the black hole and the stellar component in differentially rotating flattened systems is analysed. The interplay between velocity diffusion resulting from relaxation and black hole star accretion is investigated, together with cluster rotation, using 2D+1 (20 in space and time) Fokker–Planck numerical methods. The models can reproduce the Bahcall–Wolf solution f∝E1/4 (n∝r−7/4) inside the zone of influence of the black hole. Gravo-gyro and gravo-thermal instabilities cause the system to have a faster evolution, leading to shorter collapse times with respect to non-rotating systems. Angular momentum transport and star accretion support the development of central rotation on relaxation time-scales. We explore system dissolution as a result of mass loss in the presence of an external tidal field (e.g. for globular clusters in galaxies).
- Globular clusters are among the most congested stellar systems in the Universe. Internal dynamical evolution drives them toward states of high central density, while simultaneously concentrating the most massive stars and binary systems in their cores. As a result, these clusters are expected to be sites of frequent close encounters and physical collisions between stars and binaries, making them efficient factories for the production of interesting and observable astrophysical exotica. I describe some elements of the competition among stellar dynamics, stellar evolution, and other processes that control globular cluster dynamics, with particular emphasis on pathways that may lead to the formation of blue stragglers.
- All gravitationally bound clusters expand, due to both gas loss from their most massive members and binary heating. All are eventually disrupted tidally, either by passing molecular clouds or the gravitational potential of their host galaxies. However, their interior evolution can follow two very different paths. Only clusters of sufficiently large initial population and size undergo the combined interior contraction and exterior expansion that leads eventually to core collapse. In all other systems, core collapse is frustrated by binary heating. These clusters globally expand for their entire lives, up to the point of tidal disruption. Using a suite of direct N-body calculations, we trace the ‘collapse line’ in rv-N space that separates these two paths. Here, rv and N are the cluster's initial virial radius and population, respectively. For realistic starting radii, the dividing N-value is from 104 to over 105. We also show that there exists a minimum population, Nmin, for core collapse. Clusters with N < Nmin tidally disrupt before core collapse occurs. At the Sun's Galactocentric radius, RG = 8.5 kpc, we find Nmin ≳ 300. The minimum population scales with Galactocentric radius as $R_{\rm G}^{-9/8}$. The position of an observed cluster relative to the collapse line can be used to predict its future evolution. Using a small sample of open clusters, we find that most lie below the collapse line, and thus will never undergo core collapse. Most globular clusters, on the other hand, lie well above the line. In such a case, the cluster may or may not go through core collapse, depending on its initial size. We show how an accurate age determination can help settle this issue.
- We used high-quality images acquired with the WFC3 on board the HST to probe the blue straggler star (BSS) population of the Galactic globular cluster NGC 362. We have found two distinct sequences of BSS: this is the second case, after M 30, where such a feature has been observed. Indeed the BSS location, their extension in magnitude and color and their radial distribution within the cluster nicely resemble those observed in M 30, thus suggesting that the same interpretative scenario can be applied: the red BSS sub-population is generated by mass transfer binaries, the blue one by collisions. The discovery of four new W UMa stars, three of which lying along the red-BSS sequence, further supports this scenario. We also found that the inner portion of the density profile deviates from a King model and is well reproduced by either a mild power-law (\alpha -0.2) or a double King profile. This feature supports the hypothesis that the cluster is currently undergoing the core collapse phase. Moreover, the BSS radial distribution shows a central peak and monotonically decreases outward without any evidence of an external rising branch. This evidence is a further indication of the advanced dynamical age of NGC 362: in fact, together with M 30, NGC 362 belongs to the family of dynamically old clusters (Family III) in the "dynamical clock" classification proposed by Ferraro et al. (2012). The observational evidence presented here strengthens the possible connection between the existence of a double BSS sequence and a quite advanced dynamical status of the parent cluster.
- We study the dynamical evolution of idealized stellar systems by averaging results from many N-body simulations, each having modest numbers of stars. For isolated systems with stars of uniform mass, we discuss aspects of evolution up to the point of core collapse: relaxation and its N-dependence, the evolution of the density profile, the development of the velocity dispersion and anisotropy, and the rate of stellar escape. We find that the continuum models (gas and Fokker-Planck) agree quite accurately with N-body simulations in which N is of order of a few hundred. Small deviations from these models are present at small radii and at radii from the half-mass radius outwards. They are probably associated with binary activity and with the development of anisotropy, respectively. As expected, the N-body systems are strongly anisotropic in the outer half of the mass, while in the core the velocity distribution is isotropic to good approximation. Anisotropy has a very important influence on the rate of escape of stars. We also estimate quite reliable values for the coefficient y in the Coulomb logarithm ln(gammaN) and the conductivity coefficient C in the gas model of Lynden-Bell & Eggleton. These are gamma ~= 0.11 and C ~= 0.104, respectively.
- The evolution of star clusters is studied using N-body simulations in which the evolution of single stars and binaries is taken self-consistently into account. Initial conditions are chosen to represent relatively young Galactic open clusters, such as the Pleiades, Praesepe and the Hyades. The calculations include a realistic mass function, primordial binaries and the external potential of the parent Galaxy. Our model clusters are generally significantly flattened by the Galactic tidal field, and dissolve before deep core collapse occurs. The binary fraction decreases initially because of the destruction of soft binaries, but increases later because lower mass single stars escape more easily than the more massive binaries. At late times, the cluster core is quite rich in giants and white dwarfs. There is no evidence for preferential evaporation of old white dwarfs. On the contrary, the white dwarfs formed are likely to remain in the cluster. Stars tend to escape from the cluster through the first and second Lagrange points, in the direction of and away from the Galactic Centre. Mass segregation manifests itself in our models well within an initial relaxation time. As expected, giants and white dwarfs are much more strongly affected by mass segregation than main-sequence stars. Open clusters are dynamically rather inactive. However, the combined effects of stellar mass-loss and evaporation of stars from the cluster potential drive the dissolution of a cluster on a much shorter time-scale than if these effects are neglected. The often-used argument that a star cluster is barely older than its relaxation time and therefore cannot be dynamically evolved is clearly in error for the majority of star clusters. An observation of a blue straggler in an eccentric orbit around an unevolved star or a blue straggler of more than twice the turn-off mass might indicate past dynamical activity. We find two distinct populations of blue stragglers: those formed above the main-sequence turn-off, and those which appear as blue stragglers as the cluster's turn-off drops below the mass of the rejuvenated star.
- We study the evolution of relatively young Galactic open clusters, such as the Pleiades, Praesepe and the Hyades. The calculations include a realistic mass function, primordial binaries and the external potential of the parent Galaxy. The equations of motions of all stars are computed using the GRAPE-4 while taking the evolution of single stars and binaries into account consistently. Our model clusters are generally significantly flattened in the Galactic tidal field, and dissolve before deep core collapse occurs. At late times, the cluster core is quite rich in giants and white dwarfs. There is no evidence for preferential evaporation of old white dwarfs, on the contrary the formed white dwarfs are likely to remain in the cluster. Stars tend to escape from the cluster through the first and second Lagrange points, in the direction of and away from the Galactic center. Mass segregation manifests itself in our models well within an initial relaxation time. As expected, giants and white dwarfs are much more strongly affected by mass segregation than main-sequence stars. The combined effect of stellar mass loss and evaporation of stars from the cluster potential drives its dissolution on a much shorter time scale than if these effects are neglected. The often-used argument that a star cluster is barely older than its relaxation time and therefore cannot be dynamically evolved is clearly in error for the majority of star clusters. An observation of a blue straggler in an eccentric orbit around an unevolved star or a blue straggler of more than twice the turn off mass might indicate past dynamical activity. We find two distinct populations of blue stragglers: those formed above the main-sequence turn off, and those which appear as blue stragglers as the cluster’s turnoff drops below the mass of the rejuvenated star.
- A collisional N-body simulation using nbody5 on a single CRAY YMP processor is followed well into the post-collapse regime. This is presently one of the largest particle numbers of all such models published, but some data for an even larger N produced by using special-purpose computers have recently been presented. In contrast to previous ensemble-averaged N-body simulations the noise here is low enough just to compare this one single run with the expectations from statistical models based on the Fokker-Planck approximation. Agreement is as good as could be expected for the case of the evolution of the Lagrangian radii, radial and tangential velocity dispersions and various core quantities. We discuss briefly approximate models to understand the number and the energy of escapers and the question of gravothermal core oscillations; although the system exhibits post-collapse oscillations they turn out to be directly binary driven and we cannot prove the existence of a gravothermal expansion at this particle number. Finally in a detailed examination of the wandering of the density centre we find, in contrast to some previous studies, a clear long-time period of the order of approximately 14 half-mass crossing times.
- We describe a fully automated gravitational scattering package capable of determining cross sections and reaction rates for binary-single-star scattering, and present some applications to systems of astrophysical interest.
- I review the development of direct N-body codes at Cambridge over nearly 40 years, high- lighting the main stepping stones. The —rst code (NBODY1) was based on the simple concepts of a force polynomial combined with individual time steps, where numerical problems due to close encounters were avoided by a softened potential. Fortuitously, the elegant Kustaanheimo-Stiefel two-body regularization soon permitted small star clusters to be studied (NBODY3). Subsequent extensions to unperturbed three- body and four-body regularization proved bene—cial in dealing with multiple interactions. Investigations of larger systems became possible with the Ahmad-Cohen neighbor scheme which was used more than 20 years ago for expanding universe models of 4000 galaxies (NBODY2). Combining the neighbor scheme with the regularization procedures enabled more realistic star clusters to be considered (NBODY5). After a period of simulations with no apparent technical progress, chain regularization replaced the treatment of compact subsystems (NBODY3, NBODY5). More recently, the Hermite integration method provided a major advance and has been implemented on the special-purpose HARP computers (NBODY4) together with an alternative version for workstations and supercomputers (NBODY6). These codes also include a variety of algorithms for stellar evolution based on fast lookup functions. The treatment of primordial binaries contains efficient procedures for chaotic two-body motion as well as tidal circularization, and special attention is paid to hierarchical systems and their stability. This family of N-body codes constitutes a powerful tool for dynamical simulations which is freely available to the astronomical community, and the massive eÜort owes much to collaborators.
- Results of scattering experiments involving hard binaries with binding energies up to a few hundred times larger than the kinetic energy of the incoming field star are reported in the form of total and differential cross sections for a variety of processes. An accurate description of equal mass binary-single star scattering over a complete range of parameters is provided. The heating of star clusters through three-body processes, when stellar collisions can be ignored, as is the case for encounters involving degenerate stars, is illustrated by plotting the average amount of energy exchange between binaries and field stars as a function of binary hardness.
- It is becoming clearer that several mechanisms, difficult to distinguish, must be responsible for the blue-straggler phenomenon. It is also highly that more than one mechanism occurs even within the same cluster to produce blue-straggler stars (BSs). There is still some ambiguity about whether BSs are single or double stars, simply because of the possibility that some BSs have merged. In the youngest clusters, perhaps high rotation in single BSs provide support for internal mixing; BSs in young to intermediate-age clusters are likely to receive this mixing support from high magnetic fields; in old-disk open clusters, globulars, and perhaps dwarf galaxies, binary mass transfer, and binary merger are likely the major causes for the production of BSs, with a contribution from binary-binary collisions and coalescence. There is considerable observational evidence of the existence of binaries in these systems. Progress has been certainly made in the last 40 yr, but BSs remain an intriguing challenge.
- This paper presents the results of several direct N-body calculations of star cluster models, containing a fraction of initial binary population, without mass loss due to stellar evolution. These primordial binaries are generated with several initial mass functions for checking their influence on the dynamical evolution of clusters. Our results show that primordial binaries dominate completely the evolution of poor clusters and control it until they are ejected or disrupted; their effect is smaller for rich clusters. The quantitative behaviour seems to be dependent on the mass spectrum. Evolution of primordial binaries is examined in detail. The binary escape rate is studied and some conclusions are presented. The final product of cluster evolution, the star cluster remnant, is also discussed.
- Hydrodynamical processes in collisions between two binary stars, presumed to have formed by tidal capture, are investigated using a smoothed particle hydrodynamics code. Stellar mergers occur frequently in such events, when the binaries approach one another sufficiently closely that they could, in principle, convert substantial internal binary binding energy to external translational energy. Relative to binary-binary collisions involving point masses, hydrodynamical effects reduce the average translational energy produced by a factor of about 3. This may be sufficient to eliminate tidal-capture binaries as a viable direct energy source in star clusters, although they may still contribute indirectly through formation and stellar evolution of mergers. Multiple mergers are common, giving rise to remnants consisting of three or even all four stars. The relevance of these results to the dynamics of star clusters is discussed, along with possible implications for the hypothesis that blue stragglers form by stellar mergers.
- The semistellar nucleus of M33 was observed with high resolution surface photometry and velocity dispersion measurements in order to study its structure and to search for a central black hole. Imagery was obtained by using the DAO/CFHT HR camera. The nucleus is unresolved, and its true core radius is r(c) not greater than about 0.10 arcsec. The true central surface brightness is mu not greater than about 11.3 R mag/sq arcsec, and the central density is rho(0) not less than about than 5 x 10 exp 5 solar masses/cubic pc. The velocity dispersion, rho equal to 21 plus or minus 3 km s/exp 1, of the nucleus was measured using the Ca II infrared triplet. The mass to light ratio is small approximately less than 0.4. There is a substantial color gradient inside the 0.5 arcsec radius. These observations suggest that the nucleus contains young stars concentrated in the center. A strict limit is derived on the mass of a central black hole, implying that we can rule out the presence of a dead quasar in M33.
- Understanding the halo populations of the Milky Way impacts upon a vast landscape of stellar, Galactic and extragalactic astrophysics. Topics likely to play important roles at this meeting are introduced, including aspects of properties of the outer halo, the halo-to-disc transition, globular cluster binary stars and dynamics, chemistry, and age determinations.
- We report on the results of the CCD photometry in the B- and V- band of 8250 stars belonging to the Local Group dwarf irregular galaxy NGC 3109. Color-magnitude diagrams and luminosity functions are constructed. The overall distribution of blue and red supergiants is given. Eighteen OB associations are identified from H-alpha maps, and their color-magnitude diagrams are discussed. A catalog of previously known and newly identified H II regions is presented. The star-forming properties that we infer from these data for NGC 3109 are similar to those of other dwarf galaxies.
- N-body simulations of open clusters with several different initial mass functions (IMFs) have been performed in order to study their influence on the dynamical evolution of clusters. These simulations differ from those presented in Paper I (de la Fuente Marcos 1995) in that they consider clusters with mass loss due to stellar evolution. The results show that for all the IMFs studied the evolution of the cluster is slowed down and the initial core collapse loses importance due to an expansion of the inner regions of the cluster. We find that the total disruption time is very IMF dependent because of different numbers of massive stars and also it depends on the richness of the cluster. Some questions concerning mass loss and formation of multiple systems are discussed briefly.
- We describe some aspects of implementing star cluster simulations on HARP. The code NBODY4 employs the Hermite scheme with hierarchical block-steps for direct integration. The algorithms have been optimized for parallel processing with the eight pipeline HARP-2 delivering a peak performance of about 1.7 Gflops for N = 10 4 particles. Hard binaries are studied by KS regularization which also uses the Hermite scheme, whereas strong interactions between 3–5 particles are treated by chain regularization. Astrophysical processes modelled include mass loss by stellar evolution, two-body tidal interaction, Roche lobe mass transfer, common envelope evolution, magnetic braking and gravitational radiation. Consistent values of stellar radii and evolution type are obtained by fast look-up. A new formulation of collision outcomes yields blue stragglers and other exotic objects. Some recent results for an open cluster model are presented.
- We present the first study of the dynamical evolution of an isolated star cluster that combines a significant population of primordial binaries with the presence of a central black hole. We use equal-mass direct N-body simulations, with N ranging from 4096 to 16 384 and a primordial binary ratio of 0–10 per cent; the black hole mass is about 1 per cent of the total mass of the cluster. The evolution of the binary population is strongly influenced by the presence of the black hole, which gives the cluster a large core with a central density cusp. Starting from a variety of initial conditions (Plummer and King models), we first encounter a phase, that last approximately 10 half-mass relaxation times, in which binaries are disrupted faster compared to analogous simulations without a black hole. Subsequently, however, binary disruption slows down significantly due to the large core size. The dynamical interplay between the primordial binaries and the black hole thus introduces new features with respect to the scenarios investigated so far, where the influence of the black hole and of the binaries have been considered separately. A large core to half-mass radius ratio appears to be a promising indirect evidence for the presence of an intermediate-mass black hole in old globular clusters.
- The use of direct Fokker-Planck calculations for studying star cluster evolution is discussed. Cohn's (1978, 1979) basic algorithm for spherical systems of identical point masses and its application to the study of core collapse is reviewed. Extensions by Merritt (1981) to treat a mass spectrum and by Goodman (1983) to include strong scattering and cluster rotation are discussed. The application of this method to the study of core collapse and cluster life thereafter is reviewed. Prospects for future development of the method are discussed, emphasizing the development of physical realistic models for interpreting Hubble Space Telescope observations of globular cluster structure.
- This paper consists of two independent parts. (1) The Monte Carlo method for computing the evolution of spherical stellar systems has been modified so that the computation can be continued after the time of formation of the central singularity. Results are presented for systems with equal and unequal star masses. The initial core-halo formation is followed by a general expansion of the cluster, while the central singularity absorbs a growing fraction of the total negative energy.(2) Theoretical expressions of the ‘diffusion coefficients’, which describe the effect of encounters in a stellar system, contain a factor In(γN) where N is the number of stars and γ is a constant usually taken to be of the order of 0.4. A reconsideration of the ‘non-dominant terms’ leads to a substantially lower value, of the order of 0.15 for equal masses and 0.075 for unequal masses with a typical distribution. This correction improves the agreement between N-body and Monte Carlo simulations of spherical systems.
- The authors concentrate on the theoretical picture of core collapse and subsequent evaporation of globular clusters; this study is currently gaining momentum both from recent theoretical developments and from observations indicating that a significant fraction of globular clusters may already have completed their collapse phase and are now entering the equivalent of a main-sequence stage in stellar evolution. A theoretical discussion of the physical processes that play an important role in the pre- and postcollapse evolution of a globular cluster is given. The authors review various computational approaches to cluster evolution, summarize relevant observational methods and results, and outline directions for future theoretical and observational research.
- Interactions between binaries and other stars in the core of a collapsed model cluster are analyzed. The processes of binary formation, destruction, and hardening are discussed and compared with previous analytical predictions. It is found that, while the closest encounters retain much of their three-body character, most other interactions are significantly affected by the cluster environment. These results are interpreted in the context of a core viewed as a collection of short-lived 'clumps' of stars, rather than as a single dynamical unit.
- We present here some recent (and very preliminary) finding from a study of the stages of the post-core-collapse evolution of an isolated cluster of identical point “stars”. The method used to follow the behavior of the system is the unified N-body/statistical treatment described in detail by McMillan and Lightman (1984a) and by Lightman and McMillan elsewhere in this volume. Briefly, the method combines the standard “large-N” and “small-N” approaches to the problem in the régimes where they are appropriate by treating the inner regions (r < rN) exactly with a regularized Aarseth N-body code (Aarseth, 1972), while permitting stars at greater and greater radii to retain less and less of their individual identities, ultimately treating the outer portions of the system (r > KrN) in an almost purely statistical fashion.
- We have analyzed the effect of vectorization on several schemes of direct N-body integration. We find that an Aarseth-type code with a neighbour scheme, which is the fastest on a scalar processor, is still the fastest on vector processors for large N. For N=1000, the overall gain in speed is found to be a factor of 10 both in theory and experiment. In the presence of hard binaries, however, the gain is reduced significantly.
- A comprehensive review of the theory of galactic dynamics is presented. Key empirical facts about stellar systems are briefly reviewed, and the ingredients needed to construct galaxy models are assembled, including potential theory, stellar orbits, and the theory of the equilibrium configurations of stellar systems. The stability of these configurations and the theory of spiral structures are discussed. Collisions and encounters between stellar systems are considered, and two-body realization and the approach to statistical equilibrium in star clusters are addressed. It is shown how the observable properties of galaxies such as their luminosities and colors are changed by the aging of their constituent stellar populations. Finally, it is shown that most of the mass in the universe is locked up in some still invisible form.
- A Fokker-Planck numerical code is used to compute the evolution of an idealized globular star cluster through the phases of collapse, bounce, and reexpansion. The cluster is initially composed of 3×105stars, each of mass 0.7 M_sun;, distributed as a Plummer model with core radius 0.8 pc. The model allows for the formation of close binaries via the two-body tidal-capture process and subsequent heating and ejection phenomena, but neglects the possibility of fusion into more massive stars. Collapse is reversed at t = 15 trh when the central density has increased by a factor of 103.5, due to the rapid ejection of binaries which have come to dominate the core. During reexpansion, the core regions may undergo overstable oscillations. Most of the ≡103 close tidally captured binaries present at core collapse are formed well before and ejected slowly afterward. During late stages, binary formation and ejection are nearly in equilibrium, with the cumulative ejection reaching one-fifth of the original mass by the end of the integration.
- A gaseous model is used to study the evolution (on a relaxation time-scale) of the core of a spherical cluster of identical stars. The system is enclosed in a distant adiabatic boundary, and the energy imparted by interaction with binaries (mainly in the core) is modeled by a simple analytical form. The results for the phase of the evolution which follows the initial collapse of the core are compared with those of other authors, and the qualitative differences which appeared in previous published results are reconciled. The post-collapse behavior is sometimes steady, but sometimes exhibits a linear instability which leads to the nonlinear 'gravothermal oscillations' discovered by Sugimoto and Bettwieser. Results are presented which demonstrate the nature of these different behaviors.
- N-body simulations of open star clusters containing initial binaries and supplemental binary-binary scattering experiments have been performed in order to study the production of dynamically ejected runaway stars. These simulations differ from those presented in Paper I (Leonard and Duncan 1988) in that they consider clusters with a mass spectrum and a binary energy spectrum. It has been found that dynamically ejected runaways have a maximum velocity of ≳200 km s-1, a binary frequency of ≃10%, and a mass-velocity relation in which the lowest mass stars have the highest velocities. All three of these properties are consistent with those of the Gies and Bolton [Astrophys. J. Suppl. 61, 419 (1986)] sample of OB runaway stars. Also, the mass ratios and eccentricities of dynamically ejected binaries are consistent with those of known runaway binaries. Finally, there are enough young star clusters in the Galactic disk to account for the observed number of OB runaways. In conclusion, the dynamical ejection hypothesis appears to be a viable explanation for the OB runaway stars.
- Physical arguments are presented to show that two-body, tidal-capture binaries should form in abundance during the evolution of globular clusters by the time that core collapse begins. Interactions amongst these binaries and with core single stars will cause ejections from the cluster which pump energy into the system producing a bounce and re-expansion. Detailed numerical Fokker-Planck evolutionary calculations presented here confirm this scenario and indicate that this process is likely to be the dominant energy input for most clusters. During the reexpansion phase r (core) is proportional to the cube of t, and r(half) is proportional to t exp 2/3, with the core containing several hundred very close binary star systems.
- The production of runaway stars by the dynamical-ejection mechanism in an open star cluster containing 50 percent binaries of equal mass and energy is investigated theoretically by means of numerical simulations using the NBODY5 code of Aarseth (1985). The construction of the models is outlined, and the results are presented graphically and characterized in detail. It is shown that binary-binary collisions capable of producing runaways can occur (via formation and disruption, with some stellar collisions, of hierarchical double binaries) in clusters of relatively low density (e.g., pc-sized clusters of O or B stars). The frequency of binaries in the runaway population is found to vary between 0 and 50 percent, with the majority of runaways being unevolved early-type stars.
- N-body simulations of dynamical evolution of open clusters have been computed with the purpose of comparing them with observations. Most of the models contain 1000 bodies with masses following a power-law mass function of slope α = -2.75 and mean mass 0.5 M_sun;. Neutron stars or white dwarfs (depending on the initial stellar mass) are generated by instantaneous changes in individual masses, when stars reach the end of their main sequence life. Close approaches between particles are treated by a two-body regularization technique that allows to follow binary evolution in detail.
- To extend the search for spectroscopic binaries in the globular cluster M3, the authors have obtained more than 300 new radial velocities for the 111 giants previously observed by Gunn and Griffin (1979). For one of the stars, von Zeipel 164, four observations spanning ten years show a velocity variation with an amplitude of at least 18 km s-1 and a period of perhaps a few years. The authors believe this to be a strong candidate for the first spectroscopic binary to be found in a globular cluster.
- Evolution of a gravitational 1000-body system is calculated to analyze gravothermal oscillation in discrete particle systems. It is found that suprathermal particles are produced as a result of binary hardenings which transfer their energy to the mean field particles very slowly. Therefore, three modes share the total energy. They are: (1) the energy of the mean field, which controls the global configuration of the system; (2) the energy of correlations, whose development releases binding energies of binaries and triggers expansion; and (3) the energy of suprathermal particles, which plays the role of an additional energy reservoir acting as a buffer to avoid a sudden and great amount of energy input from the developing correlations to the mean field. Because of the moderately large number of particles, the energy transport between the core and the halo is better analyzed than in a 100-body system (Makino et al, 1986). It is concluded that the distribution of inwardly decreasing temperature and the associated heat flow towards the central core drive the gravothermal expansion quite similarly as in the gravothermal oscillation of gaseous models.
- A direct N-body simulation of a 3000-body equal-mass system to study the postcollapse evolution of globular clusters was performed. After the initial collapse, an expansion of large amplitude was observed. During this expansion the temperature profile showed a temperature inversion similar to those observed in gas models and Fokker-Planck calculations, which suggests a gravothermal origin for the expansion. Thereafter, however, only oscillatory behavior of small amplitude was observed. It may be interpreted as follows: few binaries still remained in the core, giving too high an energy generation rate for the gravothermal oscillation. In a realistic system with much larger N the effect of binaries in the core is relatively weak, so that the gravothermal oscillations may continue.
- Radial velocity measurements of giants in the globular clusters 47 Tuc, M2, M3, M 12, M 13, and M 71 were used to identify six stars that are probably binaries. These stars show radial velocity variations larger than 10 km/s and are not known to be photometric variables. Their frequency among cluster giants is 1.5 percent, which suggests that about 10 percent of all stars in the surveyed clusters are the primary of a binary. The radial distribution of the six binary candidates is consistent with that of the giants, but is less centrally concentrated than that expected for a population of objects with twice the giant mass.
- A new, general-purpose code for evolving three-dimensional, self-gravitating fluids in astrophyics, both with and without collisionless matter, is described. In this TREESPH code, hydrodynamic properties are determined using a Monte Carlo-like approach known as smoothed particle hydrodynamics (SPH). Unlike most previous implementations of SPH, gravitational forces are computed with a hierarchical tree algorithm. Multiple expansions are used to approximate the potential of distant groups of particles, reducing the cost per step. More significantly, the improvement in efficiency is achieved without the introduction of a grid. A unification of SPH with the hierarchical tree method is a natural way of allowing for larger N within a Lagrangian framework. The data structures used to manipulate the grouping of particles can be applied directly to certain aspects of the SPH calculation.
- Numerical integrations of encounters of pairs of binaries have been used to study the class of interactions, called fly-bys, in which the two-binary configuration survives. It is shown that these typically weak interactions can be treated by means of a first-order perturbation theory. A simple simulation model for obtaining the energy transfer rate between various degrees of freedom has been constructed. The model was employed to estimate the additional energy transfer arising from impact parameters larger than those used in the numerical experiments. In the hard binary limit the total energy transfer caused by binary-binary encounters is dominated by the collisional interactions in which the two-binary configuration is destroyed.
- The presence of initial binary systems in stellar clusters seems to have a dominant effect on the dynamical evolution of the whole cluster. Simulations carried out with a direct N-body method are discussed for several models with N = 300 equal-mass objects, with 20 percent of initial binaries and different binary binding energy. These calculations show that the binaries with values of binding energy (h) in the range 5-10 times the mean kinetic energy interact strongly with the field stars and among themselves. The main result of these interactions is an enhanced expansion of the core and of the whole cluster. The binary heating prevents the gravitational collapse for a time longer than the usual collapse time for systems without binaries. Energetic binaries with h greater than 25 concentrate at the center, but their binary-binary interactions produce disruption and escape of the components, with fewer consequences for cluster evolution. Binary/binary interactions occur in central regions, while single-star-binary interactions mainly occur in outer regions. The former produce core heating and are mostly responsible for the escaping stars, while the latter contribute to a strong expansion of the halo. In all the simulations there was no formation of new persistent hard binaries.
- The auhors apply their new "hybrid" stellar dynamical computer code to the gravitational collapse and post collapse evolution of a globular cluster, in the approximation of point particles of initially equal mass and for a time equal to twice the time to collapse. Among the results is the formation of a hard central binary system, reversal of core collapse and expansion due to the heat input from this binary, ejection of the binary from the core, and recollapse of the core.
- N-body calculations have shown that hard binaries play a dominant role in the evolution of small stellar systems. However, only one or two such binaries form by dynamical interactions and since the final components tend to be massive, their effect and observability in open clusters are limited to relatively short time-scales. In the present investigation we consider an initial population of binaries with physical evolution times exceeding the cluster age. These calculations may be applicable to a later phase when the heavier stars have suffered mass loss.
- The evolution of a self-gravitating N-body system is discussed. To grasp the physics clearly the system is confined in an adiabatic wall so that there are no particles escaping from the system. The authors carried out several of 100-body calculations. After the initial collapse the oscillation of the core was observed, where the expansion of the core was triggered by energy input from binaries. However, the nature of the oscillation is not clear enough in this case, because the number of particles is small and because the energy release even from a single hard binary is excessively large as compared with the total energy of the system. To limit the excessive energy release the authors also carried out calculations using a softened potential. In this case the oscillation is more clearly observed.
- Gravitational encounters of pairs of binaries have been studied numerically. Various cross-sections have been calculated for qualitative final results of the interaction and for energy transfer between the binding energy and the center of mass kinetic energy. The distribution of the kinetic energies, resulting from the gravitational collision, were found to be virtually independent of the impact velocity in the case of collision of hard binaries. It was found that one out of five collisions, which are not simple fly-by's, leads to the formation of a stable three-body system.
- To study the post-collapse evolution of clusters, where fluctuations are significant due to the number of stars in the cluster core being of the order of 100, N-body experiments with N = 1000 and 3000 were conducted. The effects of the finite size of stars are not considered, and a point-mass approximation is assumed. The analysis indicates that the core oscillates with an amplitude of the order of 10 in density (an amplitude much lower that that found in the fluid model of star clusters) and that these oscillations are directly related to binary activity. It is suggested that this phenomenon also occurs in systems with N equal to one million, and that core oscillations continue for a long time, with the period of oscillations becoming increasingly longer.
- Gravitational encounters of pairs of hard binaries with unequal energies, but equal mass stars, have been studied numerically. Special attention was drawn to the phenomenon called collision, which cannot be treated as a fly-by. A semi-analytical theory for obtaining the collision cross-section and outcome distributions was developed and the free parameters fitted to numerical observations. Good agreement between theory and experiments confirms the theoretical assumption that the class of interactions, which results in a disruption into one binary and two separately escaping stars, can be treated in terms of two independent random ejections. The energy distribution of the first ejection is approximately scaled by the harmonic mean of the binding energies of the binaries as corrected somewhat for the impact energy, while the disruption of the remaining three-body system has its total energy as the scaling factor. With increasing binding energy ratio the number of hierarchical three-body systems, resulting from collisions, becomes larger. For equal energies it is 20 percent and is 50 percent if the binding energies differ by a factor of about 4.
- The numerical Fokker-Planck determinations of core collapse in a one-component star cluster shows that a nonisothermal self-similar structure develops in the region between the shrinking isothermal core and the halo during the late stages of the core collapse. The region is characterized by the radial profiles of the stellar density, the gravitational potential, and velocity dispersion following the power laws; the central velocity dispersion increases with the central density. The data provide new evidence for the identification of the late stage of core collapse with the gravothermal instability of Lynden-Bell and Wood (1968).
- Analytical models are used to compute the evolution of the core of a stellar system due simultaneously to stellar evaporation, which causes the system (core) to contract, and to its binaries, which cause it to expand by progressively decreasing its binding energy. The evolution of the system is determined by two parameters: the initial number of stars in the system and the fraction of its stars which are binaries. For a fixed binary fraction, stellar evaporation initially dominates the dynamical evolution if the initial number is sufficiently large, due to the fact that the rate of evaporation is determined chiefly by long-range encounters which increase in importance as the number of stars in the system increases. If stellar evaporation initially dominates, the system first contracts, but as the number of remaining stars in the system decreases by evaporation, the system reaches a minimum radius and a maximum density, and then it expands monotonically as the number of remaining stars decreases further. Open clusters expand monotonically from the beginning if they have anything approaching average Population I binary frequencies. Globular clusters are highly deficient in binaries in order to have formed and retained the high-density stellar cores observed in most of them. The binary fraction for these systems is estimated to be no more than 0.15.
- A method that makes it possible to study the evolution of globular clusters from the end of the violent relaxation phase far beyond the critical moment is presented. The method describes the process of evolution and gives the rate of star escape and the energies of escaping stars and mass segregation. It is shown that the structure of the singularity appearing in the cluster center as a result of the gravothermal catastrophe has only a slight effect on subsequent cluster evolution. The evolution of a nonisolated globular cluster (mean relaxation time, 1.5 x 10 to the 9th yr) comprising stars with a continuous mass spectrum is computed over 20 x 10 to the 9th yr. Mass and shock heating are both included. It is shown that in the first 5 x 10 to the 9th yr of the cluster's evolution the mass loss retards the collapse and is essential for the energy balance.
- Gravothermal oscillations of postcollapse star clusters are investigated by studying a self-gravitating, conducting gas sphere. It is shown that self-similar solutions can be constructed if the energy generation term has an appropriate functional form. Solutions are obtained for energy generation by binaries created in three-body processes. The solutions have a single dimensionless parameter which can be related to the total number of stars N or to the central concentration. The linear stability of the solutions is studied, and it is found that they are stable for N less than about 7000, overstable for N between 7000 and 40,000, and unstable for larger N. For very large N, the instability reduces to the gravothermal instability of an infinite isothermal sphere. The amplitude of nonlinear gravothermal oscillations is estimated and found to increase with N. Implications for globular clusters and for N-body simulations are discussed.
- The evolution of globular clusters is modeled in a detailed Fokker-Planck calculation and is followed well beyond core collapse. The model includes the heating effects on the cluster caused by the dynamical formation and evolution of binaries, captured in three-body encounters and hardened by interactions with passing stars. As expected, energy generation by binaries reverses core collapse at sufficiently high central densities and drives a reexpansion of the core. The previous findings of Sugimoto and Bettwieser (1983) that this expansion is unstable to the development of large-amplitude oscillations in the central density and core radius. The implications of the results for the long-term evolution of globular clusters, are discussed.
- A theoretical framework for analyzing the computational cost of gravitational N-body codes is introduced and applied to three different types of direct-summation codes, including the type of Aarseth code which has found most general use. The method of analysis, based on the probability distribution of nearest-neighbor distances, is described. The number of time steps required for a variety of different versions of the Aarseth scheme and a variety of physical models of spherical star clusters is estimated in order to measure the effects of different degrees of central concentration. Analytical estimates of computer time required are compared with actual measurements, and the validity of the scaling outside the range actually tested is discussed. A practical result for planning star cluster simulations on the next generation of supercomputers is derived. It is found that the consumption of computer time can be very centrally concentrated.
- In the present gaseous model for the postcollapse evolution of globular clusters, the existence of a gravothermally unstable isothermal configuration with a central singularity of definite functional form is demonstrated, and the evolution towards such a configuration is computed numerically, taking energy release through the hardening of binary stars into account. The configuration is found to be approached by the cluster in an oscillatory rather than monotonic fashion. The core oscillates between high density cusp and moderate central density isothermal core states. Only a minute energy input is needed to drive this gravothermal oscillation, thereby resolving the question as to how the nonexistence of the collapsed globular cluster can be reconciled with the theoretical prediction of the central singularity.
- Observers, numerical experimenters, and theorists use a misleadingly similar language when describing star clusters, although the operational definitions of quantities such as core radius and central density differ considerably. These differences are investigated, and a class of coordinate-independent, scale-free measurements for local and global quantities, particularly suited to small N-body systems, are introduced. It is shown, by means of analytical estimates and Monte Carlo experiments, that the quantities measured are closely related to independently defined global parameters, such as the core radius and the core density. Similarities and differences between these and the definitions for corresponding quantities which are used for observations and for theoretical models, considering finite-number effects as well as those systematic discrepancies which persist in the continuum limit, are discussed. This discussion applies both to star clusters and to clusters of galaxies.
- The method of Heggie (1974) is used to derive the globally regular equations of motion for the gravitational N-body problem, employing a modified notation to render the regularized Hamiltonian more tractable. Attention is also given to alternative time transformations and some differences in formulation. The use of a modified time transformation makes possible the treatment of two simultaneous pair collisions. The method is recommended for the study of small systems, especially when it is used only temporarily, for the integration of the system over critical interactions.
- The effects of an initial binary population on the evolution of an isolated globular cluster are investigated by Monte Carlo techniques. In all models, the central regions develop a marked concentration of binaries as a result of mass stratification; in the innermost core, binaries ultimately predominate and react more with each other than with single stars. For the models considered, 42-92% of the binary energy released goes into reaction products which escape from the cluster, the percentage increasing with binary hardness. The energy imparted to the remaining cluster maintains the expansion of the system and postpones the collapse of the core. For most models, this collapse finally occurs, perhaps in part because most of the energy released is transmitted to stars outside the collapsing core in which the reactions take place. Up to 30% of the binaries present initially have been dissociated by the time of core collapse.
- Results from numerical integrations of random binary–binary encounters have been used to study the heating and mass loss of dense stellar systems by binary collisions. We used the obtained distributions and cross-sections to simulate binary collisions in a system with nearly Maxwellian velocity distribution. The loss of kinetic energy and mass by escape of reaction results was taken into account and thus a realistic estimate for the effect of these interactions was obtained. These results confirm the earlier estimates by other authors that in equal mass systems binaries must be rather numerous in order to be dynamically important. However, it is demonstrated that with increasing binary masses the importance of binary–binary reactions grows rapidly. A simple comparison of the heating rates due to binary–single star and binary–binary encounters shows that typically these are comparable or the latter dominates. It is concluded that collisions between binaries may contribute much to the dynamics of dense stellar systems provided the binaries are suitably massive.
- A unified N-body and statistical treatment of stellar dynamics is developed and applied to the late stages of core collapse and early stages of post collapse evolution in globular clusters. A 'hybrid' computer code is joined to a direct N-body code which is used to calculate exactly the behavior of particles in the inner spatial region, and the combination is used to follow particles statistically in the outer spatial region. A transition zone allows the exchange of particles and energy between the two regions. The main application results include: formation of a hard central binary system, reversal of core collapse and expansion due to the heat input from this binary, ejection of the binary from the core, and recollapse of the core; density profiles that form a one-parameter sequence during the core oscillations; and indications that these oscillations will eventually cease.
- Globular star clusters provide a unique laboratory. In astronomy they present an opportunity to study dense stellar systems that are more accessible than galactic nuclei. In statistical mechanics, concepts of negative heat capacity and resulting gravothermal instabilities challenge the present framework of statistical deceptions of dynamical systems. In computer science, modelling their evolution poses an extreme challenge to the hardware and software capabilities of the next generation of parallel computers, and provides an ideal test case for teraflop machines.
- MANYof the globular clusters in our Galaxy have probably undergone core collapse, and are currently re-expanding1,2. This re-expansion requires a central energy source. Previously proposed mechanisms are either inefficient or may produce unacceptably bright cores3. Here we explore the most conservative solution to this problem. We suggest that primordial binaries, for which there is now direct evidence4–6, could provide the necessary energy. We show that this mechanism leads to relatively large core sizes, containing ~1% of the total cluster mass. Such a cluster would have a resolvable core (with a size of the order of arcseconds) which would consist mostly of binaries.
- Letr 1,r 2,r 3 be arbitrary coordinates of the non-zero interacting mass-pointsm 1,m 2,m 3 and define the distancesR 1=|r 1–r 3|,R 2=|r 2–r 3|,R=|r 1–r 2|. An eight-dimensional regularization of the general three-body problem is given which is based on Kustaanheimo-Stiefel regularization of a single binary and possesses the properties:(i) The equations of motion are regular for the two-body collisionsR 10 orR 20. (ii) Provided thatRR 1 orRR 2, the equations of motion are numerically well behaved for close triple encounters. Although the requirementR min (R 1,R 2) may involve occasional transformations to physical variables in order to re-label the particles, all integrations are performed in regularized variables. Numerical comparisons with the standard Kustaanheimo-Stiefel regularization show that the new method gives improved accuracy per integration step at no extra computing time for a variety of examples. In addition, time reversal tests indicate that critical triple encounters may now be studied with confidence.The Hamiltonian formulation has been generalized to include the case of perturbed three-body motions and it is anticipated that this procedure will lead to further improvements ofN-body calculations.
- A method for numerically integrating the N-body gravitational problem is described. We take advantage of the fact that the force on a star can be divided into two parts which operate on different time scales. One part is due to the stars in the immediate vicinity of the star in question and another part due to the distant stars. The part of the force due to the far away stars changes much more slowly than the component due to the nearby stars. Hence that part of the force does not have to be recalculated as frequently as that due to the nearby stars. For systems with large N, most of the stars constitute the “distant” stars and the considerable saving of computing time allows us to integrate systems with up to 1000 particles.
- "Serie A, no. 3752." Thesis (doctoral)--Universite de Paris, 1961.
