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The Milky way Galaxy is brimming with free-floating objects, including stars, planets and planetesimals. For the purpose of this chapter, we define a free-floating object as a solid body that is not orbited by a considerably more massive body. A planet then is considered free floating if it is not orbiting a star but it may be orbiting another planet. A binary planet, or planet-moon pair that is not orbiting a star, is then considered free floating. Most free-floating objects are not born as such because most objects form in some sort of coordinated environmental effort, such as a star forming region or a circum-stellar disk. Free-floating stars then originate from dissolved clusters. Free floating planets are ejected from their parent star in an internal dynamical encounter with another planet or stripped from the star by other means such as a supernova or a nearby passing star. Free floating (interstellar) planetesimals probably form in a similar fashion as free-floating planets. The number of free-floating objects in the Galaxy can be large. With billions of stars and planets, and trillions of interstellar planetesimals. Although free-floating planets appear to be quite common (a few hundred have been observed), only two interstellar planetesimals have been discovered so far. The expectation, however, is that they outnumber the stars in the Galaxy by a considerable margin. We expect them to be found more frequently once large new instruments come online, such as the Vera Cooper Rubin Observatory., Comment: Preprint of a chapter for the 'Encyclopedia of Astrophysics' (Editor-in-Chief Ilya Mandel, Section Editor Dimitri Veras) to be published by Elsevier as a Reference Module

We use the stellar evolution code BPASS and the gravitational wave simulation code LEGWORK to simulate populations of compact binaries that may be detected by the in-development space-based gravitational wave (GW) detector LISA. Specifically, we simulate the Magellanic Clouds and binary populations mimicking several globular clusters, neglecting dynamical effects. We find that the Magellanic Clouds would have a handful of detectable sources each, but for globular clusters the amount of detectable sources would be less than one. We compare our results to earlier research and find that our predicted numbers are several tens of times lower than calculations using the stellar evolution code BSE that take dynamical effects into account, but also calculations using the stellar evolution code SeBa for the Magellanic Clouds. This correlates with earlier research which compared BPASS models for GW sources in the Galactic disk with BSE models and found a similarly sized discrepancy. We analyse and explain this discrepancy as being caused by differences between the stellar evolution codes, particularly in the treatment of mass transfer and common-envelope events in binaries, where in BPASS mass transfer is more likely to be stable and tends to lead to less orbital shrinkage in the common-envelope phase than in other codes. This difference results in fewer compact binaries with periods short enough to be detected by LISA existing in the BPASS population. For globular clusters, we conclude that the impact of dynamical effects is uncertain from the literature, but the differences in stellar evolution have an effect of a factor of a few tens., Comment: 11 pages, 2 figures, 4 tables. Submitted to A&A

Located at the core of the Galactic Centre, the S-star cluster serves as a remarkable illustration of chaos in dynamical systems. The long-term chaotic behaviour of this system can be studied with gravitational $N$-body simulations. By applying a small perturbation to the initial position of star S5, we can compare the evolution of this system to its unperturbed evolution. This results in the two solutions diverging exponentially, defined by the separation in position space $\delta_{r}$, with an average Lyapunov timescale of $\sim$420 yr, corresponding to the largest positive Lyapunov exponent. Even though the general trend of the chaotic evolution is governed in part by the supermassive black hole Sagittarius $\rm A^{*}$ (Sgr $\rm A^{*}$), individual differences between the stars can be noted in the behaviour of their phase-space curves. We present an analysis of the individual behaviour of the stars in this Newtonian chaotic dynamical system. The individuality of their behaviour is evident from offsets in the position space separation curves of the S-stars and the black hole. We propose that the offsets originate from the initial orbital elements of the S-stars, where Sgr $\rm A^{*}$ is considered in one of the focal points of the Keplerian orbits. Methods were considered to find a relationship between these elements and the separation in position space. Symbolic regression turns out to provide the clearest diagnostics for finding an interpretable expression for the problem. Our symbolic regression model indicates that $\left\langle\delta_r\right\rangle \propto e^{2.3}$, implying that the time-averaged individual separation in position space increases rapidly with the initial eccentricity of the S-stars., Comment: Accepted for publication in A&A

Recently we estimated that about 5 percent of supermassive black hole triple systems are fundamentally unpredictable. These gargantuan chaotic systems are able to exponentially magnify Planck length perturbations to astronomical scales within their interaction timescale. These results were obtained in the zero angular momentum limit, which we naively expected to be the most chaotic. Here, we generalise to triple systems with arbitrary angular momenta by systematically varying the initial virial ratio. We find the surprising result that increasing the angular momentum enhances the chaotic properties of triples. This is not only explained by the longer life times, allowing for a prolonged exponential growth, but also the maximum Lyapunov exponent itself increases. For the ensemble of initially virialised triple systems, we conclude that the percentage of unpredictable supermassive black hole triples increases to about 30 percent. A further increase up to about 50 percent is reached when considering triples on smaller astrophysical scales. Fundamental unpredictability is thus a generic feature of chaotic, self-gravitating triple populations., Comment: 13 pages, 11 figures, 2 tables. Submitted to MNRAS

Dynamical chaos is a fundamental manifestation of gravity in astrophysical, many-body systems. The spectrum of Lyapunov exponents quantifies the associated exponential response to small perturbations. Analytical derivations of these exponents are critical for understanding the stability and predictability of observed systems. This essay presents a new model for chaos in systems with eccentric and crossing orbits. Here, exponential divergence is not a continuous process but rather the cumulative effect of an ever-increasing linear response driven by discrete events at regular intervals, i.e., punctuated chaos. We show that long-lived systems with punctuated chaos can magnify Planck length perturbations to astronomical scales within their lifetime, rendering them fundamentally indeterministic., Comment: Essay written for the Gravity Research Foundation 2023 Awards for Essays on Gravitation (Honorable Mention). Accepted by International Journal of Modern Physics D

In self-gravitating $N$-body systems, small perturbations introduced at the start, or infinitesimal errors that are produced by the numerical integrator or are due to limited precision in the computer, grow exponentially with time. For Newton's gravity, we confirm earlier results that for relatively homogeneous systems, this rate of growth per crossing time increases with $N$ up to $N \sim 30$, but that for larger systems, the growth rate has a weaker scaling with $N$. For concentrated systems, however, the rate of exponential growth continues to scale with $N$. In relativistic self-gravitating systems, the rate of growth is almost independent of $N$. This effect, however, is only noticeable when the system's mean velocity approaches the speed of light to within three orders of magnitude. The chaotic behavior of systems with more than a dozen bodies for the usually adopted approximation of only solving the pairwise interactions in the Einstein-Infeld-Hoffmann equation of motion is qualitatively different than when the interaction terms (or cross terms) are taken into account. This result provides a strong motivation for follow-up studies on the microscopic effect of general relativity on orbital chaos, and on the influence of higher-order cross-terms in the Taylor-series expansion of the Einstein-Infeld-Hoffmann equations of motion., Comment: A&A in press

We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem. To this end, we extend the Brutus N-body code to include Post-Newtonian pairwise terms up to 2.5 order, and the first order Taylor expansion to the Einstein-Infeld-Hoffmann equations of motion. The degree to which our system is relativistic depends on the scaling of the total mass (the unit size was 1 parsec). Using the Brutus method of convergence, we test for time-reversibility in the conservative regime, and demonstrate that we are able to obtain definitive solutions to the relativistic three-body problem. It is also confirmed that the minimal required numerical accuracy for a successful time-reversibility test correlates with the amplification factor of an initial perturbation. When we take into account dissipative effects through gravitational wave emission, we find that the duration of the resonance, and the amount of exponential growth of small perturbations depend on the mass scaling. For a unit mass <= 10 MSun, the system behavior is indistinguishable from the Newtonian case, and the resonance always ends in a binary and one escaping body. For a mass scaling up to 1e7 MSun, relativity gradually becomes more prominent, but the majority of the systems still dissolve. The first mergers start to appear for a mass of ~1e5 MSun, and between 1e7 MSun and 1e9 MSun all systems end prematurely in a merger. These mergers are preceded by a gravitational wave driven in-spiral. For a mass scaling >= 1e9 MSun, all systems result in a gravitational wave merger upon the first close encounter. Relativistic three-body encounters thus provide an efficient pathway for resolving the final parsec problem. The onset of mergers at the characteristic mass scale of 1e7 MSun potentially leaves an imprint in the mass function of supermassive black holes., Comment: Accepted by Phys Rev D. 10 pages, 6 figures

Context. As the importance of Gravitational Wave (GW) Astrophysics increases rapidly, astronomers in different fields and with different backgrounds can have the need to get a quick idea of which GW source populations can be detected by which detectors and with what measurement uncertainties. Aims. The GW-Toolbox is an easy-to-use, flexible tool to simulate observations on the GW universe with different detectors, including ground-based interferometers (advanced LIGO, advanced VIRGO, KAGRA, Einstein Telescope, and also customised designs), space-borne interferometers (LISA and a customised design), pulsar timing arrays mimicking the current working ones (EPTA, PPTA, NANOGrav, IPTA) and future ones. We include a broad range of sources such as mergers of stellar mass compact objects, namely black holes, neutron stars and black hole-neutron stars; and supermassive black hole binaries mergers and inspirals, Galactic double white dwarfs in ultra-compact orbit, extreme mass ratio inspirals and Stochastic GW backgrounds. Methods. We collect methods to simulate source populations and determine their detectability with the various detectors. The paper aims at giving a comprehensive description on the algorithm and functionality of the GW-Toolbox. Results. The GW-Toolbox produces results that are consistent with more detailed calculations of the different source classes and can be accessed with a website interface (gw-universe.org) or as a python package (https://bitbucket.org/radboudradiolab/gwtoolbox). In the future, it will be upgraded with more functionality., Comment: 21 pages, 26 figures, 4 tables. Submitted to Astronomy & Astrophysics; Website url: gw-universe.org; comments on v2: made some revisions according to referee's comments; added Cosmic Explorer; comments on v3: make some more clarification in the caveats section, on the waveform and uncertainty estimation

The (yet-to-be confirmed) discovery of a Neptune-sized moon around the ~3.2 Jupiter-mass planet in Kepler 1625 puts interesting constraints on the formation of the system. In particular, the relatively wide orbit of the moon around the planet, at ~40 planetary radii, is hard to reconcile with planet formation theories. We demonstrate that the observed characteristics of the system can be explained from the tidal capture of a secondary planet in the young system. After a quick phase of tidal circularization, the lunar orbit, initially much tighter than 40 planetary radii, subsequently gradually widened due to tidal synchronization of the spin of the planet with the orbit, resulting in a synchronous planet-moon system. Interestingly, in our scenario the captured object was originally a Neptune-like planet, turned into a moon by its capture., Comment: Accepted for publication in ApJL. 7 pages, 5 figures

The Oort cloud (OC) probably formed more than 4$\,$Gyr ago and has been moving with the Sun in the Galaxy since, exposed to external influences, most prominently to the Galactic tide and passing field stars. Theories suggest that other stars might posses exocomets distributed similarly to our OC. We study the erosion of the OC and the possibility for capturing exocomets during the encounters with such field stars. We carry out simulations of flybys, where both stars are surrounded by a cloud of comets. We measure how many exocomets are transferred to the OC, how many OC's comets are lost, and how this depends on the other star's mass, velocity and impact parameter. Exocomets are transferred to the OC only during relatively slow ($\lesssim0.5\,$km$\,$s$^{-1}$) and close ($\lesssim10^5\,$AU) flybys and these are expected to be extremely rare. Assuming that all passing stars are surrounded by a cloud of exocomets, we derive that the fraction of exocomets in the OC has been about $10^{-5}$--$10^{-4}$. Finally we simulate the OC for the whole lifetime of the Sun, taking into account the encounters and the tidal effects. The OC has lost 25--65% of its mass, mainly due to stellar encounters, and at most 10% (and usually much less) of its mass can be captured. However, exocomets are often lost shortly after the encounter that delivers them, due to the Galactic tide and consecutive encounters., Comment: Accepted for publication in MNRAS. 16 pages, 10 figures

Most stars form in star clusters and stellar associated. To understand the roles of star cluster environments in shaping the dynamical evolution of planetary systems, we carry out direct $N$-body simulations of four planetary systems models in three different star cluster environments with respectively N=2k, 8k and 32k stars. In each cluster, an ensemble of initially identical planetary systems are assigned to solar-type stars with $\sim 1 M_{\odot}$ and evolved for 50~Myr. We found that following the depletion of protoplanetary disks, external perturbations and planet-planet interactions are two driving mechanisms responsible for the destabilization of planetary systems. The planet survival rate varies from $\sim 95\%$ in the N=2k cluster to $\sim 60\%$ in the N=32k cluster, which suggests that most planetary systems can indeed survive in low-mass clusters, except in the central regions. We also find that planet ejections through stellar encounters are cumulative processes, as only $\sim 3\%$ of encounters are strong enough to excite the eccentricity by $\Delta e \geq 0.5$. Short-period planets can be perturbed through orbit crossings with long-period planets. When taking into account planet-planet interactions, the planet ejection rate nearly doubles, and therefore multiplicity contributes to the vulnerability of planetary systems. In each ensemble, $\sim 0.2\%$ of planetary orbits become retrograde due to random directions of stellar encounters. Our results predict that young low-mass star clusters are promising sites for next-generation planet surveys, yet low planet detection rates are expected in dense globular clusters such as 47 Tuc. Nevertheless, planets in denser stellar environments are likely to have shorter orbital periods, which enhances their detectability., Comment: 19 pages, 13 figures, 4 tables, accepted for publication in MNRAS

We show that black-hole High-Mass X-ray Binaries (HMXBs) with O- or B-type donor stars and relatively short orbital periods, of order one week to several months may survive spiral in, to then form Wolf-Rayet (WR) X-ray binaries with orbital periods of order a day to a few days; while in systems where the compact star is a neutron star, HMXBs with these orbital periods never survive spiral-in. We therefore predict that WR X-ray binaries can only harbor black holes. The reason why black-hole HMXBs with these orbital periods may survive spiral in is: the combination of a radiative envelope of the donor star, and a high mass of the compact star. In this case, when the donor begins to overflow its Roche lobe, the systems are able to spiral in slowly with stable Roche-lobe overflow, as is shown by the system SS433. In this case the transferred mass is ejected from the vicinity of the compact star (so-called "isotropic re-emission" mass loss mode, or "SS433-like mass loss"), leading to gradual spiral-in. If the mass ratio of donor and black hole is $>3.5$, these systems will go into CE evolution and are less likely to survive. If they survive, they produce WR X-ray binaries with orbital periods of a few hours to one day. Several of the well-known WR+O binaries in our Galaxy and the Magellanic Clouds, with orbital periods in the range between a week and several months, are expected to evolve into close WR-Black-Hole binaries,which may later produce close double black holes. The galactic formation rate of double black holes resulting from such systems is still uncertain, as it depends on several poorly known factors in this evolutionary picture. It might possibly be as high as $\sim 10^{-5}$ per year., Comment: MNRAS in press

Approximately $0.2 \pm 0.2$ of white dwarfs (WDs) show signs of pollution by metals, which is likely due to the accretion of tidally disrupted planetary material. Models invoking planet-planet interactions after WD formation generally cannot explain pollution at cooling times of several Gyr. We consider a scenario in which a planet is perturbed by Lidov-Kozai oscillations induced by a binary companion and exacerbated by stellar mass loss, explaining pollution at long cooling times. Our computed accretion rates are consistent with observations assuming planetary masses between $\sim 0.01$ and $1\,M_\mathrm{Mars}$, although nongravitational effects may already be important for masses $\lesssim 0.3 \, M_\mathrm{Mars}$. The fraction of polluted WDs in our simulations, $\sim 0.05$, is consistent with observations of WDs with intermediate cooling times between $\sim 0.1$ and 1 Gyr. For cooling times $\lesssim 0.1$ Gyr and $\gtrsim 1$ Gyr, our scenario cannot explain the high observed pollution fractions of up to 0.7. Nevertheless, our results motivate searches for companions around polluted WDs., Comment: Accepted for publication in MNRAS Letters. 5 pages, 5 figures

Hot Jupiters (HJs) are Jupiter-like planets that reside very closely to their host star, within $\sim 0.1\,\mathrm{AU}$. Their formation is not well understood. It is generally believed that they cannot have formed in situ, implying that some form of migration must have occurred after their initial formation. We study the production of HJs through secular evolution in multiplanet systems with three to five planets. In this variant of high-$e$ migration, the eccentricity of the orbit of the innermost planet is excited on secular time-scales, triggering orbital migration due to tidal dissipation. We use a secular dynamics code and carry out a population synthesis study. We find that HJs are only produced if the viscous time-scale is short ($\approx 0.014$ yr). In contrast, in up to $\approx 0.3$ of systems, the innermost planet is tidally disrupted. The orbital period distribution is peaked around 5 d, consistent with observations. The median HJ mass is $1\,M_\mathrm{J}$ with a maximum of $\approx 2 \, M_\mathrm{J}$, similar to observed HJs. Approximately 0.1 of the HJs have retrograde orbits with respect to the stellar spin. We do not find any warm Jupiters in our simulations, i.e. planets with semimajor axes between 0.1 and 1 AU., Comment: Revised to match MNRAS publication. 14 pages, 10 figures

According to Munoz-Gutierrez et al. (2015) the orbit of comet 1P/Halley is chaotic with a surprisingly small Lyapunov time scale of order its orbital period. In this work we analyse the origin of chaos in Halley's orbit and the growth of perturbations, in order to get a better understanding of this unusually short time scale. We perform N-body simulations to model Halley's orbit in the Solar System and measure the separation between neighbouring trajectories. To be able to interpret the numerical results, we use a semi-analytical map to demonstrate different growth modes, i.e. linear, oscillatory or exponential, and transitions between these modes. We find the Lyapunov time scale of Halley's orbit to be of order 300 years, which is significantly longer than previous estimates in the literature. This discrepancy could be due to the different methods used to measure the Lyapunov time scale. A surprising result is that next to Jupiter, also encounters with Venus contribute to the exponential growth in the next 3000 years. Finally, we note an interesting application of the sub-linear, oscillatory growth mode to an ensemble of bodies moving through the Solar System. Whereas in the absence of encounters with a third body the ensemble spreads out linearly in time, the accumulation of weak encounters can increase the lifetime of such systems due to the oscillatory behaviour., Comment: Accepted for MNRAS, 10 pages, 6 figures

We present a method for studying the secular gravitational dynamics of hierarchical multiple systems consisting of nested binaries, which is valid for an arbitrary number of bodies and arbitrary hierarchical structure. We derive the Hamiltonian of the system and expand it in terms of the -- assumed to be -- small ratios $x_i$ of binary separations. At the lowest nontrivial expansion order (quadrupole order, second order in $x_i$), the Hamiltonian consists of terms which, individually, depend on binary pairs. At higher orders, in addition to terms depending on binary pairs, we also find terms which, individually, depend on more than two binaries. In general, at order $n$ in $x_i$, individual terms depend on at most $n-1$ binaries. We explicitly derive the Hamiltonian including all terms up and including third order in $x_i$ (octupole order), and including the binary pairwise terms up and including fifth order in $x_i$. These terms are orbit averaged, and we present a new algorithm for efficiently solving the equations of motion. This algorithm is highly suitable for studying the secular evolution of hierarchical systems with complex hierarchies, making long-term integrations of such systems feasible. We show that accurate results are obtained for multiplanet systems with semimajor axis ratios as large as $\approx 0.4$, provided that high-order terms are included. In addition to multiplanet systems with a single star, we apply our results to multistar systems with multiple planets., Comment: Revised to match MNRAS publication. 48 pages, 36 figures

Transiting circumbinary planets are more easily detected around short-period than long-period binaries, but none have yet been observed by {\it Kepler} orbiting binaries with periods shorter than seven days. In triple systems, secular Kozai-Lidov cycles and tidal friction (KLCTF) have been shown to reduce the inner orbital period from $\sim 10^4$ to a few days. Indeed, the majority of short-period binaries are observed to possess a third stellar companion. Using secular evolution analysis and population synthesis, we show that KLCTF makes it unlikely for circumbinary transiting planets to exist around short-period binaries. We find the following outcomes. (1) Sufficiently massive planets in tight and/or coplanar orbits around the inner binary can quench the KL evolution because they induce precession in the inner binary. The KLCTF process does not take place, preventing the formation of a short-period binary. (2) Secular evolution is not quenched and it drives the planetary orbit into a high eccentricity, giving rise to an unstable configuration, in which the planet is most likely ejected from the system. (3) Secular evolution is not quenched but the planet survives the KLCTF evolution. Its orbit is likely to be much wider than the currently observed inner binary orbit, and is likely to be eccentric and inclined with respect to the inner binary. These outcomes lead to two main conclusions: (1) it is unlikely to find a massive planet on a tight and coplanar orbit around a short-period binary, and (2) the properties of circumbinary planets in short-period binaries are constrained by secular evolution., Comment: Revised to match MNRAS publication. 24 pages, 22 figures

We study the secular gravitational dynamics of quadruple systems consisting of a hierarchical triple system orbited by a fourth body. These systems can be decomposed into three binary systems with increasing semimajor axes, binaries A, B and C. The Hamiltonian of the system is expanded in ratios of the three binary separations, and orbit-averaged. Subsequently, we numerically solve the equations of motion. We study highly hierarchical systems that are well described by the lowest-order terms in the Hamiltonian. We find that the qualitative behaviour is determined by the ratio $\mathcal{R}_0$ of the initial Kozai-Lidov (KL) time-scales of the binary pairs AB and BC. If $\mathcal{R}_0\ll 1$, binaries AB remain coplanar if this is initially the case, and KL eccentricity oscillations in binary B are efficiently quenched. If $\mathcal{R}_0\gg 1$, binaries AB become inclined, even if initially coplanar. However, there are no induced KL eccentricity oscillations in binary A. Lastly, if $\mathcal{R}_0\sim 1$, complex KL eccentricity oscillations can occur in binary A that are coupled with the KL eccentricity oscillations in B. Even if binaries A and B are initially coplanar, the induced inclination can result in very high eccentricity oscillations in binary A. These extreme eccentricities could have significant implications for strong interactions such as tidal interactions, gravitational wave dissipation, and collisions and mergers of stars and compact objects. As an example, we apply our results to a planet+moon system orbiting a central star, which in turn is orbited by a distant and inclined stellar companion or planet, and to observed stellar quadruples., Comment: Revised to match MNRAS publication. 27 pages, 16 figures

Flares in X-ray and near infrared are observed above the quiescent emission of the supermassive black hole (SBH) in the Galactic Centre (GC) at a rate of approximately once per day. One proposed energy source for these flares is the tidal disruption of planetesimals with radius $\gtrsim 10$ km passing within $\sim$ 1 AU of the SBH. Very little is known about the formation and evolution of planetesimals in galactic nuclei such as the GC, making predictions for flaring event rates uncertain. We explore two scenarios for the formation of planetesimals in the GC: (1) in a large-scale cloud bound to the SBH, and (2) in debris discs around stars. We model their orbital evolution around the SBH using the Fokker-Planck equation and investigate the effect of gravitational interactions with various relevant perturbers. Our predicted flaring rate, $\approx 0.6 \, \mathrm{day^{-1}}$, is nearly independent of the distribution of perturbers. Moreover, it is insensitive to scenarios (1) or (2). The assumed number of planetesimals per star is consistent with debris discs around stars in the Solar neighbourhood. In scenario (1) this implies that the number of planetesimals formed in the large-scale cloud is strongly correlated with the number of stars, and this requires finetuning for our results to be consistent with the observed flaring rate. We favour the alternative explanation that planetesimals in the GC are formed in debris discs around stars, similar to the Solar neighbourhood., Comment: Accepted for publication in MNRAS. 21 pages, 12 figures

We present a novel method for efficient direct integration of gravitational N-body systems with a large variation in characteristic time scales. The method is based on a recursive and adaptive partitioning of the system based on the connected components of the graph generated by the particle distribution combined with an interaction-specific time step criterion. It uses an explicit and approximately time-symmetric time step criterion, and conserves linear and angular momentum to machine precision. In numerical tests on astrophysically relevant setups, the method compares favourably to both alternative Hamiltonian-splitting integrators as well as recently developed block time step-based GPU-accelerated Hermite codes. Our reference implementation is incorporated in the HUAYNO code, which is freely available as a part of the AMUSE framework., Comment: 12 pages, 8 figures, accepted for A&A

General relativistic precession limits the ability of gravitational encounters to increase the eccentricity $e$ of orbits near a supermassive black hole (SBH). This "Schwarzschild barrier" (SB) has been shown to play an important role in the orbital evolution of stars like the galactic center S-stars. However, the evolution of orbits below the SB, $e>e_\mathrm{SB}$, is not well understood; the main current limitation is the computational complexity of detailed simulations. Here we present an $N$-body algorithm that allows us to efficiently integrate orbits of test stars around a SBH including general relativistic corrections to the equations of motion and interactions with a large ($\gtrsim 10^3$) number of field stars. We apply our algorithm to the S-stars and extract diffusion coefficients describing the evolution in angular momentum $L$. We identify three angular momentum regimes, in which the diffusion coefficients depend in functionally different ways on $L$. Regimes of lowest and highest $L$ are well-described in terms of non-resonant relaxation (NRR) and resonant relaxation (RR), respectively. In addition, we find a new regime of "anomalous relaxation" (AR). We present analytic expressions, in terms of physical parameters, that describe the diffusion coefficients in all three regimes, and propose a new, empirical criterion for the location of the SB in terms of the $L$-dependence of the diffusion coefficients. Subsequently we apply our results to obtain the steady-state distribution of angular momentum for orbits near a SBH., Comment: Accepted for publication in MNRAS. 34 pages, 32 figures

We review the implementation of individual particle time-stepping for N-body dynamics. We present a class of integrators derived from second order Hamiltonian splitting. In contrast to the usual implementation of individual time-stepping, these integrators are momentum conserving and show excellent energy conservation in conjunction with a symmetrized time step criterion. We use an explicit but approximate formula for the time symmetrization that is compatible with the use of individual time steps. No iterative scheme is necessary. We implement these ideas in the HUAYNO (available online at www.amusecode.org) code and present tests of the integrators and show that the presented integration schemes shows good energy conservation, with little or no systematic drift, while conserving momentum and angular momentum to machine precision for long term integrations., Comment: 20 pages, 9 figures, accepted for publication in New Astronomy

Young star clusters like R136 in the Large Magellanic Cloud and NGC 3603, Westerlund 1, and 2 in the Milky Way are dynamically more evolved than expected based on their current relaxation times. In particular, the combination of a high degree of mass segregation, a relatively low central density, and the large number of massive runaway stars in their vicinity are hard to explain with the monolithic formation of these clusters. Young star clusters can achieve such a mature dynamical state if they formed through the mergers of a number of less massive clusters. The shorter relaxation times of less massive clusters cause them to dynamically evolve further by the time they merge, and the merger product preserves the memory of the dynamical evolution of its constituent clusters. With a series of $N$-body simulations, we study the dynamical evolution of single massive clusters and those that are assembled through merging smaller clusters together. We find that the formation of massive star clusters through the mergers of smaller clusters can reproduce the currently observed spatial distribution of massive stars, the density, and the characteristics (number and mass distribution) of the stars ejected as runaways from young dense clusters. We therefore conclude that these clusters and possibly other young massive star clusters formed through the mergers of smaller clusters., Comment: 13 pages, 13 figures, accepted for publication in ApJ

(Abridged) Interacting galaxies are well-known for their high star formation rates and rich star cluster populations, but the rapidly changing tidal field can also efficiently destroy clusters. We use numerical simulations of merging disc galaxies to investigate which mechanism dominates. The simulations include a model for the formation and dynamical disruption of the entire star cluster population. We find that the dynamical heating of clusters by tidal shocks is about an order of magnitude higher in interacting galaxies than in isolated galaxies. This is driven by the increased gas density, and is sufficient to destroy star clusters at a higher rate than new clusters are formed: the total number of clusters in the merger remnant is 2-50% of the amount in the progenitor discs, with low-mass clusters being disrupted preferentially. By adopting observationally motivated selection criteria, we find that the observed surplus of star clusters in nearby merging galaxies is caused by the bias to detect young, massive clusters. We provide a general expression for the survival fraction of clusters, which increases with the gas depletion time-scale. Due to the preferential disruption of low-mass clusters, the mass distribution of the surviving star clusters in a merger remnant develops a peak at a mass of about 10^3 Msun, which evolves to higher masses at a rate of 0.3-0.4 dex per Gyr. The peak mass initially depends weakly on the galactocentric radius, but this correlation disappears as the system ages. We discuss the similarities between the cluster populations of the simulated merger remnants and (young) globular cluster systems. Our results suggest that the combination of cluster formation and destruction should be widespread in the dense star-forming environments at high redshifts, which could provide a natural origin to present-day globular cluster systems., Comment: 15 pages, 9 figures, 2 tables; Accepted for publication in MNRAS. A movie of the full time sequence in Figure 1 can be found at http://www.mpa-garching.mpg.de/~diederik/1m11clusters.html

(Abridged) The formation and evolution of star cluster populations are related to the galactic environment. Cluster formation is governed by processes acting on galactic scales, and star cluster disruption is driven by the tidal field. In this paper, we present a self-consistent model for the formation and evolution of star cluster populations, for which we combine an N-body/SPH galaxy evolution code with semi-analytic models for star cluster evolution. The model includes star formation, feedback, stellar evolution, and star cluster disruption by two-body relaxation and tidal shocks. We apply the model by simulating a suite of 9 isolated disc galaxies and 24 galaxy mergers. The evolutionary histories of individual clusters in these simulations are discussed to illustrate how the environment of clusters changes in time and space. The resulting variability of the disruption rate with time and space affects the properties of star cluster populations. The combined effect of clusters escaping their dense formation sites (`cluster migration') and the preferential disruption of clusters residing in dense environments (`natural selection') implies that the mean disruption rate of the population decreases with cluster age. This affects the slope of the cluster age distribution, which becomes a function of the star formation rate density. The evolutionary histories of clusters in a galaxy merger vary widely and determine which clusters survive the merger. This impacts the age distributions and the locations of the surviving clusters at all times during a merger. We conclude that accounting for the interplay between the formation, disruption, and orbital histories of clusters enables a more sophisticated interpretation of observed properties of cluster populations, thereby extending the role of cluster populations as tracers of galaxy evolution., Comment: 28 pages, 18 figures, 2 tables; Accepted for publication in MNRAS

Observations and theory both suggest that star clusters form sub-virial (cool) with highly sub-structured distributions. We perform a large ensemble of N-body simulations of moderate-sized (N=1000) cool, fractal clusters to investigate their early dynamical evolution. We find that cool, clumpy clusters dynamically mass segregate on a short timescale, that Trapezium-like massive higher-order multiples are commonly formed, and that massive stars are often ejected from clusters with velocities > 10 km/s (c.f. the average escape velocity of 2.5 km/s). The properties of clusters also change rapidly on very short timescales. Young clusters may also undergo core collapse events, in which a dense core containing massive stars is hardened due to energy losses to a halo of lower-mass stars. Such events can blow young clusters apart with no need for gas expulsion. The warmer and less substructured a cluster is initially, the less extreme its evolution., Comment: Accepted for publication in MNRAS; supplementary material can be downloaded at http://sgoodwin.staff.shef.ac.uk/all_plots.pdf.gz

We describe the results of a search for the remnants of the Sun's birth cluster among stars in the Hipparcos Catalogue. This search is based on the predicted phase space distribution of the Sun's siblings from simple simulations of the orbits of the cluster stars in a smooth Galactic potential. For stars within 100 pc the simulations show that it is interesting to examine those that have small space motions relative to the Sun. From amongst the candidate siblings thus selected there are six stars with ages consistent with that of the Sun. Considering their radial velocities and abundances only one potential candidate, HIP 21158, remains but essentially the result of the search is negative. This is consistent with predictions by Portegies Zwart (2009) on the number of siblings near the Sun. We discuss the steps that should be taken in anticipation of the data from the Gaia mission in order to conduct fruitful searches for the Sun's siblings in the future., Comment: 7 pages, 9 figures, accepted for publication in MNRAS

The relation between the globular cluster luminosity function (GCLF, dN/dlogL) and globular cluster mass function (GCMF, dN/dlogM) is considered. Due to low-mass star depletion, dissolving GCs have mass-to-light (M/L) ratios that are lower than expected from their metallicities. This has been shown to lead to an M/L ratio that increases with GC mass and luminosity. We model the GCLF and GCMF and show that the power law slopes inherently differ (1.0 versus 0.7, respectively) when accounting for the variability of M/L. The observed GCLF is found to be consistent with a Schechter-type initial cluster mass function and a mass-dependent mass-loss rate., Comment: 4 pages, 2 figures. To appear in the proceedings of "Galaxy Wars: Stellar Populations and Star Formation in Interacting Galaxies" (Tennessee, July 2009)

The conversion of the globular cluster luminosity function (GCLF, dN/dlogL) to the globular cluster mass function (GCMF, dN/dlogM) is addressed. Dissolving globular clusters (GCs) become preferentially depleted in low-mass stars, which have a high mass-to-light ratio. This has been shown to result in a mass-to-light ratio (M/L) that increases with GC luminosity or mass, because more massive GCs have lost a smaller fraction of their stars than low-mass GCs. Using GC models, we study the influence of the luminosity dependency of M/L on the inferred GCMF. The observed GCLF is consistent with a powerlaw or Schechter type GC initial mass function in combination with a cluster mass-dependent mass loss rate. Below the peak, the logarithmic slope of the GCMF is shallower than that of the GCLF (0.7 versus 1.0), whereas the peak mass is 0.1-0.3 dex lower when accounting for the variability of M/L than in the case where a constant M/L is adopted., Comment: 6 pages, 5 figures; accepted for publication in ApJ Letters

We present a new method to detect and quantify mass segregation in star clusters. It compares the minimum spanning tree (MST) of massive stars with that of random stars. If mass segregation is present, the MST length of the most massive stars will be shorter than that of random stars. This difference can be quantified (with an associated significance) to measure the degree of mass segregation. We test the method on simulated clusters in both 2D and 3D and show that the method works as expected. We apply the method to the Orion Nebula Cluster (ONC) and show that the method is able to detect the mass segregation in the Trapezium with a `mass segregation ratio' \Lambda_{MSR}=8.0 \pm 3.5 (where \Lambda_{MSR}=1 is no mass segregation) down to 16 \Msun, and also that the ONC is mass segregated at a lower level (~2.0 \pm 0.5) down to 5 \Msun. Below 5 \Msun we find no evidence for any further mass segregation in the ONC., Comment: Accepted in MNRAS

We reconstruct the initial two-body relaxation time at the half mass radius for a sample of young $\aplt 300$ Myr star clusters in the large Magellanic cloud. We achieve this by simulating star clusters with 12288 to 131072 stars using direct $N$-body integration. The equations of motion of all stars are calculated with high precision direct $N$-body simulations which include the effects of the evolution of single stars and binaries. We find that the initial relaxation times of the sample of observed clusters in the large Magellanic cloud ranges from about 200 Myr to about 2 Gyr. The reconstructed initial half-mass relaxation times for these clusters has a much narrower distribution than the currently observed distribution, which ranges over more than two orders of magnitude., Comment: To appear in ``Mass loss from stars and the evolution of stellar clusters''. Proc. of a workshop held in honour of H.J.G.L.M. Lamers, Lunteren, The Netherlands. Eds. A. de Koter, L. Smith and R. Waters (San Francisco: ASP)

Some young star clusters show a degree of mass segregation that is inconsistent with the effects of standard two-body relaxation from an initially unsegregated system without substructure, in virial equilibrium, and it is unclear whether current cluster formation models can account for this degree of initial segregation in clusters of significant mass. In this Letter we demonstrate that mergers of small clumps that are either initially mass segregated, or in which mass segregation can be produced by two-body relaxation before they merge, generically lead to larger systems which inherit the progenitor clumps' segregation. We conclude that clusters formed in this way are naturally mass segregated, accounting for the anomalous observations and suggesting that this process of prompt mass segregation due to initial clumping should be taken fully into account in constructing cluster dynamical models., Comment: 13 pages, 4 figures, submitted to ApJL

The observed average lifetime of the population of star clusters in the Solar Neighbourhood, the Small Magellanic Cloud and in selected regions of M51 and M33 is compared with simple theoretical predictions and with the results of N-body simulations. The empirically derived lifetimes (or disruption times) of star clusters depend on their initial mass as t_dis ~ Mcl^0.60 in all four galaxies. N-body simulations have shown that the predicted disruption time of clusters in a tidal field scales as t_dis^pred ~ t_rh^0.75 t_cr^0.25, where t_rh is the initial half-mass relaxation time and t_cr is the crossing time for a cluster in equilibrium. We show that this can be approximated accurately by t_dis^pred ~ M_cl^0.62 for clusters in the mass range of about 10^3 to 10^6 M_sun, in excellent agreement with the observations. Observations of clusters in different extragalactic environments show that t_dis also depends on the ambient density in the galaxies where the clusters reside. Linear analysis predicts that the disruption time will depend on the ambient density of the cluster environment as t_dis ~ rho_amb^-0.5. This relation is consistent with N-body simulations., Comment: 7 pages, 4 figures. Accepted for publication in A&A

The nature of ultraluminous X-ray sources (ULXs) is presently unknown. A possible explanation is that they are accreting intermediate mass black holes (IBHs) that are fed by Roche lobe overflow from a tidally captured stellar companion. We show that a star can circularize around an IBH without being destroyed by tidal heating (in contrast to the case of M_bh> 10^6 M_sun massive black holes in galactic centers, where survival is unlikely). We find that the capture and circularization rate is of the order of 5 \times 10^-8 yr^-1, almost independently of the cluster's relaxation time. We follow the luminosity evolution of the binary system during the main sequence Roche lobe overflow phase and show it can maintain ULX-like luminosities for >10 Myr. In particular, we show that the ULX in the young cluster MGG-11 in star-burst galaxy M82, which possibly harbors an IBH, is well explained by this mechanism, and we predict that \gtrsim 10% of similar clusters with IBHs have a tidally captured circularized star. The cluster can evaporate on a time-scale shorter than the lifetime of the binary. This raises the possibility of a ULX that outlives its host cluster, or even lights up only after the cluster has evaporated, in agreement with observations of host-less ULXs., Comment: Accepted version (ApJL), new figure, typos corrected

We study the efficiency at which a black hole or dense star cluster spirals-in to the Galactic centre. This process takes place on a dynamical friction time scale, which depends on the value of the Coulomb logarithm ln(L). We determine the accurate value of this parameter using the direct N-body method, a tree algorithm and a particle-mesh technique with up to 2 million plus one particles. The three different techniques are in excellent agreement. Our result for the Coulomb logarithm appears to be independent of the number of particles. We conclude that ln(L) = 6.6 +/- 0.6 for a massive point particle in the inner few parsec of the Galactic bulge. For an extended object, like a dense star cluster, ln(L) is smaller, with a value of the logarithm argument L inversely proportional to the object size., Comment: 11 pages, 12 figures, MNRAS, in press revised version following referee's comments, references updated, typos corrected

We study the growth rate of stars via stellar collisions in dense star clusters, calibrating our analytic calculations with direct N-body simulations of up to 65536 stars, performed on the GRAPE family of special-purpose computers. We find that star clusters with initial half-mass relaxation times of about 25 Myr are dominated by stellar collisions, the first collisions occurring at or near the point of core collapse, which is driven by the segregation of the most massive stars to the cluster center, where they end up in hard binaries. The majority of collisions occur with the same star, resulting in the runaway growth of a supermassive object. This object can grow up to about 0.1% of the mass of the entire star cluster and could manifest itself as an intermediate-mass black hole (IMBH). The phase of runaway growth lasts until mass loss by stellar evolution arrests core collapse. Star clusters older than about 5 Myr and with present-day half-mass relaxation times less than 100 Myr are expected to contain an IMBH., Comment: 19 pages, ApJ in press

We study the occurrence of physical collisions between stars in young and compact star cluster. The calculations are performed on the GRAPE-4 with the starlab software environment which include the dynamical evolution and the nuclear evolution of all stars and binaries. The selection of the initial conditions is based on existing and well observed star clusters, such as R136 in the 30 Doradus region in the Large Magellanic Cloud and the Arches and Quintuplet star clusters in the vicinity of the Galactic center. Collisions between stars occurred rather frequently in our models. At any time a single star dominates the collision history of the system. The collision rate of this runaway merger scales with the initial relaxation time of the cluster and is independent on other cluster parameters, such as the initial mass function or the initial density profile of the cluster. Subsequent encounters result in a steady grow in mass of the coagulating star, until it escapes or explodes in a supernova. The collision rate in these models is about 0.00022 collisions per star per Myr for a cluster with an initial relaxation time of 1 Myr., Comment: 15 pages, to be published in 'Stellar Collisions, Mergers, and their Consequences', ASP Conference Series (ed. M. Shara)

We present a new model for the Galactic population of close double white dwarfs. The model accounts for the suggestion of the avoidance of a substantial spiral-in during mass transfer between a giant and a main-sequence star of comparable mass and for detailed cooling models. It agrees well with the observations of the local sample of white dwarfs if the initial binary fraction is close to 50% and an ad hoc assumption is made that white dwarfs with mass less than about 0.3 solar mass cool faster than the models suggest. About 1000 white dwarfs brighter than V=15 have to be surveyed for detection of a pair which has total mass greater than the Chandrasekhar mass and will merge within 10 Gyr., Comment: 15 pages, 7 figures, to appear in Proc. ``The influence of binaries on stellar population studies'', Brussels, August 2000 (Kluwer, D. Vanbeveren ed.)

We model the population of double white dwarfs in the Galaxy and find a better agreement with observations compared to earlier studies, due to two modifications. The first is the treatment of the first phase of unstable mass transfer and the second the modelling of the cooling of the white dwarfs. A satisfactory agreement with observations of the local sample of white dwarfs is achieved if we assume that the initial binary fraction is ~ 50% and that the lowest mass white dwarfs (M < 0.3 Msun) cool faster than the most recently published cooling models predict. With this model we find a Galactic birth rate of close double white dwarfs of 0.05 yr^{-1}, a birth rate of AM CVn systems of 0.005 yr^{-1}, a merger rate of pairs with a combined mass exceeding the Chandrasekhar limit (which may be progenitors of SNe Ia) of 0.003 yr^{-1} and a formation rate of planetary nebulae of 1 yr^{-1}. We estimate the total number of double white dwarfs in the Galaxy as 2.5 10^8. In an observable sample with a limiting magnitude V_lim = 15 we predict the presence of ~855 white dwarfs of which ~220 are close pairs. Of these 10 are double CO white dwarfs of which one has a combined mass exceeding the Chandrasekhar limit and will merge within a Hubble time., Comment: Accepted for publication in Astronomy & Astrophysics

We study the evolution and observability of young and compact star clusters near the Galactic center, such as the Arches cluster and the Quintuplet. The star clusters are modeled with a combination of techniques; using direct N-body, integration to calculate the motions of all stars and detailed stellar and binary evolution to follow the evolution of the stars. The modeled star clusters dissolve within 10 to 60 million years in the tidal field of the Galaxy. The projected stellar density in the modeled clusters drops within 5% to 70% of the lifetime to a level comparable to the projected background density towards the Galactic center. And it will be very hard to distinguished these clusters at later age among the background stars. This effect is more severe for clusters at larger distance from the Galactic center but in projection at the same distance. Based on these arguments we conclude that the Galactic center easily hides 10 to 40 clusters with characteristics similar to the Arches and the Quintuplet cluster., Comment: 7 pages, with 2 postscript figures, to appear in Dynamics of Star Clusters and the Milky Way, eds. R. Spurzem et al. (ASP Conference Series)

The unique core-mass - radius relation for giants with degenerate helium cores enables us to reconstruct the evolution of three observed double helium white dwarfs with known masses of both components. The last mass transfer phase in their evolution must have been a spiral-in. In the formalism proposed by Webbink (1984), we can constrain the efficiency of the deposition of orbital energy into the envelope to be $1 \la \alpha \la 6$, for an envelope structure parameter $\lambda=0.5$. We find that the two standard mass transfer types (stable mass transfer and spiral-in) are both unable to explain the first phase of mass transfer for these three binaries. We use a parametric approach to describe mass transfer in low-mass binaries, where both stars are of comparable mass and find that the orbital characteristics of the observed double helium white dwarfs can be well reproduced if the envelope of the primary is lost with ~1.5 times the specific angular momentum of the initial binary. In this case no substantial spiral-in occurs., Comment: 8 pages, accepted for publication in A&A

The recent finding by Chevalier & Ilovaisky (1998) that OB-supergiant X-ray binaries have relatively large runaway velocities whereas Be/X-ray binaries have low runaway velocities, provides confirmation of the current models for the formation of these two types of systems. These predict a difference in runaway velocity of an order of magnitude. This difference basically results from the variation of the fractional helium core mass as a function of stellar mass, in combination with the conservation of orbital angular momentum during the mass transfer phase that preceded the formation of the compact object in the system. This combination results into: (i) Systematically narrower pre-supernova orbits in the OB-supergiant systems than in the Be-systems, and (ii) A larger fractional amount of mass ejected in the supernovae in high-mass systems relative to systems of lower mass. Regardless of possible kick velocities imparted to neutron stars at birth, this combination leads to a considerable difference in average runaway velocity between these two groups. The observed low runaway velocities of the Be/X-ray binaries confirm that in most cases not more than 1 to 2Msun was ejected in the supernovae that produced their neutron stars. This, in combination with the --on average-- large orbital eccentricities of these systems, indicates that their neutron stars must have received a velocity kick in the range 60 - 250 km/s at birth., Comment: reduced abstract, 13 pages, accepted by A&A

The evolution of star clusters is studied using N-body simulations in which the evolution of single stars and binaries are 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 in the Galactic tidal field, and dissolve before deep core collapse occurs. The binary fraction decreases initially due to 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 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. Open clusters are dynamically rather inactive. However, the combined effect of stellar mass loss and evaporation of stars from the cluster potential drives its dissolution on a much shorter timescale 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., Comment: reduced abstract, 33 pages (three separate color .jpg figures), submitted to MNRAS

We perform binary population synthesis calculations to study the origin and the characteristics of runaway O and B stars which are ejected by the supernova explosion of the companion star in a binary system. The number of OB runaways can be explained from supernova ejections only if: high mass stars are preferentially formed in binaries, the initial mass ratio distribution is strongly peaked to unity and stars are rejuvenated to zero age upon accretion of mass from a companion star. Taking these requirements into consideration we conclude that at most 30% of the runaway O stars but possibly all runaway B stars obtain high velocities due to supernovae in evolving binaries. Stars which obtain high velocities via supernova ejections have the following characteristics: 1) at least 10% of the high velocity B stars and half the O stars have a mass greater than the turn off mass of the cluster in which they are born and would be observed as blue stragglers in the parent cluster, 2) their equatorial rotational velocities are proportional to their space velocity and 3) between 20% and 40% of the runaways have neutron star companions but less than 1% are visible as radio pulsars in part of the orbit., Comment: 23 pages, submitted to ApJ

We consider gravitational waves emitted by various populations of compact binaries at cosmological distances. We use population synthesis models to characterize the properties of double neutron stars, double black holes and double white dwarf binaries as well as white dwarf-neutron star, white dwarf-black hole and black hole-neutron star systems. We use the observationally determined cosmic star formation history to reconstruct the redshift distribution of these sources and their merging rate evolution. The gravitational signals emitted by each source during its early-inspiral phase add randomly to produce a stochastic background in the low frequency band with spectral strain amplitude between 10^{-18} Hz^{-1/2} and 5 10^{-17} Hz^{-1/2} at frequencies in the interval [5 10^{-6}-5 10^{-5}] Hz. The overall signal which, at frequencies above 10^{-4}Hz, is largely dominated by double white dwarf systems, might be detectable with LISA in the frequency range [1-10] mHz and acts like a confusion limited noise component which might limit the LISA sensitivity at frequencies above 1 mHz., Comment: 14 pages, 14 figures, uses mn.sty, submitted to MNRAS

We investigate the evolution of globular clusters using N-body calculations and anisotropic Fokker-Planck (FP) calculations. The models include a mass spectrum, mass loss due to stellar evolution, and the tidal field of the parent galaxy. Recent N-body calculations have revealed a serious discrepancy between the results of N-body calculations and isotropic FP calculations. The main reason for the discrepancy is an oversimplified treatment of the tidal field employed in the isotropic FP models. In this paper we perform a series of calculations with anisotropic FP models with a better treatment of the tidal boundary and compare these with N-body calculations. The new tidal boundary condition in our FP model includes one free parameter. We find that a single value of this parameter gives satisfactory agreement between the N-body and FP models over a wide range of initial conditions. Using the improved FP model, we carry out an extensive survey of the evolution of globular clusters over a wide range of initial conditions varying the slope of the mass function, the central concentration, and the relaxation time. The evolution of clusters is followed up to the moment of core collapse or the disruption of the clusters in the tidal field of the parent galaxy. In general, our model clusters, calculated with the anisotropic FP model with the improved treatment for the tidal boundary, live longer than isotropic models. The difference in the lifetime between the isotropic and anisotropic models is particularly large when the effect of mass loss via stellar evolution is rather significant. On the other hand the difference is small for relaxation- dominated clusters which initially have steep mass functions and high central concentrations., Comment: 36 pages, 11 figures, LaTeX; added figures and tables; accepted by ApJ

The evolution of the comoving cosmic merger-rate density of neutron star binaries n_c(z) is calculated using a distribution of their merging times provided by population-synthesis computations of binary stars. We adopt an exponential law for the star formation rate with various timescales for different morphological types of galaxies. For elliptical galaxies also an initial burst of star formation, lasting one Gyr, is considered. The resulting n_c(z) of most models agree with the form n_c(z) \propto (1+z)^{1.5-2} for z < 2, which has been proposed for the source population of gamma-ray bursts. Assuming a standard candle luminosity, the computed brightness distribution is consistent with the BATSE results if bursts at the peak flux threshold, P = 0.4 photons cm^{-2}/s, are located at a limiting redshift of 1.9 to 3.3. Progenitors of the systems producing gamma-ray bursts at small redshift (bright) are likely to host in spiral galaxies and star forming regions whereas these at high redshift (dim) reside mainly in elliptical galaxies. The location of a burst may be up to a Mpc away from the host galaxy., Comment: 4 pages letter including 4 figures, Latex; A&A, Letters in press

We discuss the formation and evolution of binaries which contain neutron stars or black holes. It is shown that in a stellar system which for $10^{10}$ yr had star formation rate similar to the current one in the Galactic disc, the rate of neutron star mergers is $\sim 2 \times 10^{-5} yr{-1}$, consistent with the observational estimates. The rate of black hole and neutron star mergers is by an order of magnitude lower. The pairs of black holes form but do not merge because heavy mass loss by their progenitors efficiently moves the binary components apart., Comment: 9 pages, 3 figures, LaTex (uses art10.sty, frontier.sty, psfig.sty). To appear in the proceedings of the "Second Workshop on Gravitational Wave Data Analysis", held 13-15 November 1997, Orsay, France

The formation and evolution of binaries which contain two neutron stars or a neutron star with a black hole are discussed in detail. The evolution of the distributions in orbital period and eccentricity for neutron star binaries are studied as a function of time. In the model which fits the observations of high mass binary pulsars best the deposition of orbital energy into common envelopes has to be very efficient and a kick velocity distribution has to contain a significant contribution of low velocity kicks. The estimated age of the population has to be between several 100 Myr and 1 Gyr. The birthrate of binary neutron stars is about 3.4 10^{-5} per year (assuming 100% binarity) and their merger rate is about 2 x 10^{-5} per year. The merger rate of neutron star binaries is consistent with the estimated rate of gamma-ray bursts, if the latter are beamed into an opening angle of a few degrees. We argue that PSR B2303+46 is possibly formed in a scenario in which the common envelope is avoided while for the other three known high-mass binary pulsars a common envelope is required to explain their orbital period., Comment: 18 pages including 9 figures, Latex; Accepted for publication in A&A