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 fou
nd 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. For this purpose, we add some analysis tools in order to make the study less subjective. 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 a major upgrade of a Monte Carlo code which has previously been used for many studies of dense star clusters. We outline the steps needed in order to calibrate the results of the new Monte Carlo code against $N$-body simulations for large
$N$ systems, up to $N=200000$. The new version of the Monte Carlo code (called MOCCA), in addition to the features of the old version, incorporates the direct Fewbody integrator (Fregeau et al. 2004) for three- and four-body interactions, and a new treatment of the escape process based on Fukushige & Heggie (2000). Now stars which fulfil the escape criterion are not removed immediately, but can stay in the system for a certain time which depends on the excess of the energy of a star above the escape energy. They are called potential escapers. With the addition of the Fewbody integrator the code can follow all interaction channels which are important for the rate of creation of various types of objects observed in star clusters, and ensures that the energy generation by binaries is treated in a manner similar to the $N$-body model. There are at most three new parameters which have to be adjusted against $N$-body simulations for large $N$: two (or one, depending on the chosen approach) connected with the escape process, and one responsible for the determination of the interaction probabilities. The values adopted for the free parameters have at most a weak dependence on $N$. They allow MOCCA to reproduce $N$-body results with reasonable precision, not only for the rate of cluster evolution and the cluster mass distribution, but also for the detailed distributions of mass and binding energy of binaries. Additionally, the code can follow the rate of formation of blue stragglers and black hole - black hole binaries.
We investigate the epicyclic motion of stars escaping from star clusters. Using streaklines, we visualise the path of escaping stars and show how epicyclic motion leads to over- and underdensities in tidal tails of star clusters moving on circular an
d eccentric orbits about a galaxy. Additionally, we investigate the effect of the cluster mass on the tidal tails, by showing that their structure is better matched when the perturbing effect of the cluster mass is included. By adjusting streaklines to results of N-body computations we can accurately and quickly reproduce all observed substructure, especially the streaky features often found in simulations which may be interpreted in observations as multiple tidal tails. Hence, we can rule out tidal shocks as the origin of such substructures. Finally, from the adjusted streakline parameters we can verify that for the star clusters we studied escape mainly happens from the tidal radius of the cluster, given by x_L = (GM/(Omega^2-partial^2Phi/partial R^2))^{1/3}. We find, however, that there is another limiting radius, the edge radius, which gives the smallest radius from which a star can escape during one cluster orbit about the galaxy. For eccentric cluster orbits the edge radius shrinks with increasing orbital eccentricity (for fixed apocentric distance) but is always significantly larger than the respective perigalactic tidal radius. In fact, the edge radii of the clusters we investigated, which are extended and tidally filling, agree well with their (fitted) King radii, which may indicate a fundamental connection between these two quantities.
The discovery of dynamical friction was Chandrasekhars best known contribution to the theory of stellar dynamics, but his work ranged from the few-body problem to the limit of large N (in effect, galaxies). Much of this work was summarised in the tex
t Principles of Stellar Dynamics (Chandrasekhar 1942, 1960), which ranges from a precise calculation of the time of relaxation, through a long analysis of galaxy models, to the behaviour of star clusters in tidal fields. The later edition also includes the work on dynamical friction and related issues. In this review we focus on progress in the collisional aspects of these problems, i.e. those where few-body interactions play a dominant role, and so we omit further discussion of galaxy dynamics. But we try to link Chandrasekhars fundamental discoveries in collisional problems with the progress that has been made in the 50 years since the publication of the enlarged edition.
The evolution of globular clusters due to 2-body relaxation results in an outward flow of energy and at some stage all clusters need a central energy source to sustain their evolution. Henon provided the insight that we do not need to know the detail
s of the energy production in order to understand the relaxation-driven evolution of the cluster, at least outside the core. He provided two self-similar solutions for the evolution of clusters based on the view that the cluster as a whole determines the amount of energy that is produced in the core: steady expansion for isolated clusters, and homologous contraction for clusters evaporating in a tidal field. We combine these models: the half-mass radius increases during the first half of the evolution, and decreases in the second half; while the escape rate approaches a constant value set by the tidal field. We refer to these phases as `expansion dominated and `evaporation dominated. These simple analytical solutions immediately allow us to construct evolutionary tracks and isochrones in terms of cluster half-mass density, cluster mass and galacto-centric radius. From a comparison to the Milky Way globular clusters we find that roughly 1/3 of them are in the second, evaporation-dominated phase and for these clusters the density inside the half-mass radius varies with the galactocentric distance R as rho_h ~ 1/R^2. The remaining 2/3 are still in the first, expansion-dominated phase and their isochrones follow the environment-independent scaling rho_h ~ M^2; that is, a constant relaxation time-scale. We find substantial agreement between Milky Way globular cluster parameters and the isochrones, which suggests that there is, as Henon suggested, a balance between the flow of energy and the central energy production for almost all globular clusters.
Black holes and neutron stars present extreme forms of matter that cannot be created as such in a laboratory on Earth. Instead, we have to observe and analyze the experiments that are ongoing in the Universe. The most telling observations of black ho
les and neutron stars come from dense stellar systems, where stars are crowded close enough to each other to undergo frequent interactions. It is the interplay between black holes, neutron stars and other objects in a dense environment that allows us to use observations to draw firm conclusions about the properties of these extreme forms of matter, through comparisons with simulations. The art of modeling dense stellar systems through computer simulations forms the main topic of this review.
