ALBERT

Description: Accurate direct N -body simulations help to obtain detailed information about the dynamical evolution of star clusters. They also enable comparisons with analytical models and Fokker-Planck or Monte Carlo methods. nbody6 is a well-known direct N -body code for star clusters, and nbody6++ is the extended version designed for large particle number simulations by supercomputers. We present nbody6++gpu , an optimized version of nbody6++ with hybrid parallelization methods (MPI, GPU, OpenMP, and AVX/SSE) to accelerate large direct N -body simulations, and in particular to solve the million-body problem. We discuss the new features of the nbody6++gpu code, benchmarks, as well as the first results from a simulation of a realistic globular cluster initially containing a million particles. For million-body simulations, nbody6++gpu is 400–2000 times faster than nbody6 with 320 CPU cores and 32 NVIDIA K20X GPUs. With this computing cluster specification, the simulations of million-body globular clusters including 5 per cent primordial binaries require about an hour per half-mass crossing time.

Description: The formation and evolution of protoplanetary systems, the breeding grounds of planet formation, is a complex dynamical problem that involves many orders of magnitudes. To serve this purpose, we present a new hybrid algorithm that combines a Fokker–Planck approach with the advantages of a pure direct-summation N -body scheme, with a very accurate integration of close encounters for the orbital evolution of the larger bodies with a statistical model, envisaged to simulate the very large number of smaller planetesimals in the disc. Direct-summation techniques have been historically developed for the study of dense stellar systems such as open and globular clusters and, within some limits imposed by the number of stars, of galactic nuclei. The number of modifications to adapt direct-summation N -body techniques to planetary dynamics is not undemanding and requires modifications. These include the way close encounters are treated, as well as the selection process for the ‘neighbour radius’ of the particles and the extended Hermite scheme, used for the very first time in this work, as well as the implementation of a central potential, drag forces and the adjustment of the regularization treatment. For the statistical description of the planetesimal disc, we employ a Fokker–Planck approach. We include dynamical friction, high- and low-speed encounters, the role of distant encounters as well as gas and collisional damping and then generalize the model to inhomogenous discs. We then describe the combination of the two techniques to address the whole problem of planetesimal dynamics in a realistic way via a transition mass to integrate the evolution of the particles according to their masses.

Description: The formation and evolution of protoplanetary discs remains a challenge from both a theoretical and numerical standpoint. In this work, we first perform a series of tests of our new hybrid algorithm presented in Glaschke, Amaro-Seoane and Spurzem (henceforth Paper I ) that combines the advantages of high accuracy of direct-summation N -body methods with a statistical description for the planetesimal disc based on Fokker–Planck techniques. We then address the formation of planets, with a focus on the formation of protoplanets out of planetesimals. We find that the evolution of the system is driven by encounters as well as direct collisions and requires a careful modelling of the evolution of the velocity dispersion and the size distribution over a large range of sizes. The simulations show no termination of the protoplanetary accretion due to gap formation, since the distribution of the planetesimals is only subjected to small fluctuations. We also show that these features are weakly correlated with the positions of the protoplanets. The exploration of different impact strengths indicates that fragmentation mainly controls the overall mass-loss, which is less pronounced during the early runaway growth. We prove that the fragmentation in combination with the effective removal of collisional fragments by gas drag sets an universal upper limit of the protoplanetary mass as a function of the distance to the host star, which we refer to as the mill condition .

Description: We examine the effect of an accretion disc on the orbits of stars in the central star cluster surrounding a central massive black hole by performing a suite of 39 high-accuracy direct N -body simulations using state-of-the art software and accelerator hardware, with particle numbers up to 128k. The primary focus is on the accretion rate of stars by the black hole (equivalent to their tidal disruption rate for black holes in the small to medium mass range) and the eccentricity distribution of these stars. Our simulations vary not only the particle number, but disc model (two models examined), spatial resolution at the centre (characterized by the numerical accretion radius) and softening length. The large parameter range and physically realistic modelling allow us for the first time to confidently extrapolate these results to real galactic centres. While in a real galactic centre both particle number and accretion radius differ by a few orders of magnitude from our models, which are constrained by numerical capability, we find that the stellar accretion rate converges for models with N ≥ 32k. The eccentricity distribution of accreted stars, however, does not converge. We find that there are two competing effects at work when improving the resolution: larger particle number leads to a smaller fraction of stars accreted on nearly circular orbits, while higher spatial resolution increases this fraction. We scale our simulations to some nearby galaxies and find that the expected boost in stellar accretion (or tidal disruption, which could be observed as X-ray flares) in the presence of a gas disc is about a factor of 10. Even with this boost, the accretion of mass from stars is still a factor of ~100 slower than the accretion of gas from the disc. Thus, it seems accretion of stars is not a major contributor to black hole mass growth.

Description: Introducing the dragon simulation project, we present direct N -body simulations of four massive globular clusters (GCs) with 10 6 stars and 5 per cent primordial binaries at a high level of accuracy and realism. The GC evolution is computed with nbody6++gpu and follows the dynamical and stellar evolution of individual stars and binaries, kicks of neutron stars and black holes (BHs), and the effect of a tidal field. We investigate the evolution of the luminous (stellar) and dark (faint stars and stellar remnants) GC components and create mock observations of the simulations (i.e. photometry, colour–magnitude diagrams, surface brightness and velocity dispersion profiles). By connecting internal processes to observable features, we highlight the formation of a long-lived ‘dark’ nuclear subsystem made of BHs, which results in a two-component structure. The inner core is dominated by the BH subsystem and experiences a core-collapse phase within the first Gyr. It can be detected in the stellar (luminous) line-of-sight velocity dispersion profiles. The outer extended core – commonly observed in the (luminous) surface brightness profiles – shows no collapse features and is continuously expanding. We demonstrate how a King model fit to observed clusters might help identify the presence of post core-collapse BH subsystems. For global observables like core and half-mass radii, the direct simulations agree well with Monte Carlo models. Variations in the initial mass function can result in significantly different GC properties (e.g. density distributions) driven by varying amounts of early mass-loss and the number of forming BHs.

Description: We present the first detailed comparison between million-body globular cluster simulations computed with a Hénon-type Monte Carlo code, cmc , and a direct N -body code, nbody6++gpu . Both simulations start from an identical cluster model with 10 6 particles, and include all of the relevant physics needed to treat the system in a highly realistic way. With the two codes ‘frozen’ (no fine-tuning of any free parameters or internal algorithms of the codes) we find good agreement in the overall evolution of the two models. Furthermore, we find that in both models, large numbers of stellar-mass black holes (〉1000) are retained for 12 Gyr. Thus, the very accurate direct N -body approach confirms recent predictions that black holes can be retained in present-day, old globular clusters. We find only minor disagreements between the two models and attribute these to the small- N dynamics driving the evolution of the cluster core for which the Monte Carlo assumptions are less ideal. Based on the overwhelming general agreement between the two models computed using these vastly different techniques, we conclude that our Monte Carlo approach, which is more approximate, but dramatically faster compared to the direct N -body, is capable of producing an accurate description of the long-term evolution of massive globular clusters even when the clusters contain large populations of stellar-mass black holes.

Description: Dense stellar systems such as globular clusters and dense nuclear clusters are the breeding ground of sources of gravitational waves for the advanced detectors LIGO and Virgo. The stellar densities reached in these systems lead to the dynamical formation of binaries at a rate superior to what one can expect in regions of the galaxy with lower densities. Hence, these systems deserve a close study to estimate rates and parameter distribution. This is not an easy task, since the evolution of a dense stellar cluster involves the integration of N bodies with high resolution in time and space and including hard binaries and their encounters and, in the case of gravitational waves, one needs to take into account important relativistic corrections. In this work, we present the first implementation of the effect of spin in mergers in a direct-summation code, nbody6 . We employ non-spinning post-Newtonian (PN) corrections to the Newtonian accelerations up to 3.5 PN order as well as the spin–orbit coupling up to next-to-lowest order and the lowest order spin–spin coupling. We integrate spin precession and add a consistent treatment of mergers. We analyse the implementation by running a set of two-body experiments and then we run a set of 500 simulations of a stellar cluster with a velocity dispersion set to a high value to induce relativistic mergers to set a proving ground of the implementation. In spite of the large number of mergers in our tests, the application of the algorithm is robust. We find in particular the formation of a runaway black hole (BH) whose spin decays with the mass it wins, independently of the initial value of the spins of the BHs. We compare the result with 500 Monte Carlo realizations of the scenario and confirm the evolution observed with our direct-summation integrator. More remarkably, the subset of compact objects that does not undergo many mergers, and hence represent a more realistic system, has a correlation between the final absolute spin and the initial choice for the initial distribution, which could provide us with information about the evolution of spins in dense clusters once the first detections have started.

Description: The majority of stars form in star clusters and many are thought to have planetary companions. We demonstrate that multiplanet systems are prone to instabilities as a result of frequent stellar encounters in these star clusters much more than single-planet systems. The cumulative effect of close and distant encounters on these planetary systems are investigated using Monte Carlo scattering experiments. We consider two types of planetary configurations orbiting Sun-like stars: (i) five Jupiter-mass planets in the semimajor axis range 1–42 au orbiting a Solar mass star, with orbits that are initially coplanar, circular and separated by 10 mutual Hill radii and (ii) the four gas giants of our Solar system. We find that in the equal-mass planet model, 70 per cent of the planets with initial semimajor axes a  〉 40 au are either ejected or have collided with the central star or another planet within the lifetime of a typical cluster, and that more than 50 per cent of all planets with a  〈 10 au remain bound to the system. Planets with short orbital periods are not directly affected by encountering stars. However, secular evolution of perturbed systems may result in the ejection of the innermost planets or in physical collisions of the innermost planets with the host star, up to many thousands of years after a stellar encounter. The simulations of the Solar system-like systems indicate that Saturn, Uranus and Neptune are affected by both direct interactions with encountering stars, as well as planet–planet scattering. Jupiter, on the other hand, is almost only affected by direct encounters with neighbouring stars, as its mass is too large to be substantially perturbed by the other three planets. Our results indicate that stellar encounters can account for the apparent scarcity of exoplanets in star clusters, not only for those on wide-orbit that are directly affected by stellar encounters, but also planets close to the star which can disappear long after a stellar encounter has perturbed the planetary system.

Description: To understand the effects of the initial rotation on the evolution of the tidally limited clusters with mass spectrum, we have performed N -body simulations of the clusters with different initial rotations and compared the results with those of the Fokker–Planck (FP) simulations. We confirmed that the cluster evolution is accelerated by not only the initial rotation but also the mass spectrum. For the slowly rotating models, the time evolutions of mass, energy and angular momentum show good agreements between N -body and FP simulations. On the other hand, for the rapidly rotating models, there are significant differences between these two approaches at the early stage of the evolutions because of the development of bar instability in N -body simulations. The shape of the cluster for N -body simulations becomes tri-axial or even prolate, which cannot be produced by the two-dimensional FP simulations. The total angular momentum and the total mass of the cluster decrease rapidly while bar-like structure persists. After the rotational energy becomes smaller than the critical value for the bar instability, the shape of the cluster becomes nearly axisymmetric again, and follows the evolutionary track predicted by the FP equation. We have confirmed again that the energy equipartition is not completely achieved when M 2 / M 1 ( m 2 / m 1 ) 3/2  〉 0.16. By examining the angular momentum at each mass component, we found that the exchange of angular momentum between different mass components occurs, similar to the energy exchange leading to the equipartition.