N-body Simulations Of Star Clusters Research Articles

The dragon-II simulations – I. Evolution of single and binary compact objects in star clusters with up to 1 million stars

ABSTRACT We present the first results of the Dragon-II simulations, a suite of 19 N-body simulations of star clusters with up to 106 stars, with up to 33 per cent of them initially paired in binaries. In this work, we describe the main evolution of the clusters and their compact objects (COs). All Dragon-II clusters form in their centre a black hole (BH) subsystem with a density 10−100 times larger than the stellar density, with the cluster core containing $50{-}80~{{\rm per\ cent}}$ of the whole BH population. In all models, the BH average mass steeply decreases as a consequence of BH burning, reaching values 〈mBH〉 < 15 M⊙ within 10−30 relaxation times. Generally, our clusters retain only BHs lighter than 30 M⊙ over 30 relaxation times. Looser clusters retain a higher binary fraction, because in such environments binaries are less likely disrupted by dynamical encounters. We find that BH–main-sequence star binaries have properties similar to recently observed systems. Double CO binaries (DCOBs) ejected from the cluster exhibit larger mass ratios and heavier primary masses than ejected binaries hosting single CO binaries (SCOBs). Ejected SCOBs have BH masses mBH = 3−20 M⊙, definitely lower than those in DCOBs (mBH = 10−100 M⊙).

Revisiting the common envelope evolution in binary stars: A new semianalytic model for N -body and population synthesis codes

We present a novel way of modeling common envelope evolution in binary and few-body systems. We consider the common envelope inspiral as driven by a drag force with a power-law dependence in relative distance and velocity. The orbital motion is resolved either by direct N-body integration or by solving the set of differential equations for the orbital elements as derived using perturbation theory. Our formalism can model the eccentricity during the common envelope inspiral, and it gives results consistent with smoothed particles hydrodynamical simulations. We apply our formalism to common envelope events from binary population synthesis models and find that the final eccentricity distribution resembles the observed distribution of post-common-envelope binaries. Our model can be used for time-resolved common-envelope evolution in population synthesis calculations or as part of binary interactions in direct N-body simulations of star clusters.

Quantifying kinematic substructure in star-forming regions with statistical tests of spatial autocorrelation

ABSTRACT We investigate whether spatial-kinematic substructure in young star-forming regions can be quantified using Moran’s I statistic. Its presence in young star clusters would provide an indication that the system formed from initially substructured conditions, as expected by the hierarchical model of star cluster formation, even if the cluster were spatially smooth and centrally concentrated. Its absence, on the other hand, would be evidence that star clusters form monolithically. The Moran’s I statistic is applied to N-body simulations of star clusters with different primordial spatial-velocity structures, and its evolution over time is studied. It is found that this statistic can be used to reliably quantify spatial-kinematic substructure, and can be used to provide evidence as to whether the spatial-kinematic structure of regions with ages ≲6 Myr is best reproduced by the hierarchical or monolithic models of star formation. Moran’s I statistic is also able to conclusively say whether the data are not consistent with initial conditions that lack kinematic substructure, such as the monolithic model, in regions with ages up to, and potentially beyond, 10 Myrs. This can therefore provide a kinematic signature of the star cluster formation process that is observable for many Myr after any initial spatial structure has been erased.

Compact object mergers in hierarchical triples from low-mass young star clusters

Abstract A binary star orbited by an outer companion constitutes a hierarchical triple system. The outer body may excite the eccentricity of the inner binary through the von Zeipel-Lidov-Kozai (ZLK) mechanism, triggering the gravitational wave (GW) coalescence of the inner binary when its members are compact objects. Here, we study a sample of hierarchical triples with an inner black hole (BH) – BH binary, BH – neutron star (NS) binary, and BH – white dwarf (WD) binary, formed via dynamical interactions in low-mass young star clusters. Our sample of triples was obtained self-consistently from direct N-body simulations of star clusters which included up-to-date stellar evolution. We find that the inner binaries in our triples cannot merge via GW radiation alone, and the ZLK mechanism is essential to trigger their coalescence. Contrary to binaries assembled dynamically in young star clusters, binary BHs merging in triples have preferentially low mass ratios (q ≃ 0.3) and higher primary masses ($m_{\rm p} \gtrsim 40 {\, \rm M_\odot }$). We derive a local merger rate density of 0.60, 0.11 and $0.5 {-1} \, \rm Gpc^{-3}$ for BH-BH, BH-NS and BH-WD binaries, respectively. Additionally, we find that merging binaries have high eccentricities across the GW spectrum, including the LIGO-Virgo-KAGRA (LVK), LISA, and DECIGO frequencies. About 7 per cent of BH-BH and 60 per cent of BH-NS binaries will have detectable eccentricities in the LVK band. Our results indicate that the eccentricity and the mass spectrum of merging binaries are the strongest features for the identification of GW mergers from triples.

Fast Multipole Methods for N-body Simulations of Collisional Star Systems

Abstract Direct N-body simulations of star clusters are accurate but expensive, largely due to the numerous  ( N 2 ) pairwise force calculations. To solve the post-million-body problem, it will be necessary to use approximate force solvers, such as tree codes. In this work, we adapt a tree-based, optimized Fast Multipole Method (FMM) to the collisional N-body problem. The use of a rotation-accelerated translation operator and an error-controlled cell opening criterion leads to a code that can be tuned to arbitrary accuracy. We demonstrate that our code, Taichi, can be as accurate as direct summation when N > 104. This opens up the possibility of performing large-N, star-by-star simulations of massive stellar clusters and would permit large parameter space studies that would require years with the current generation of direct summation codes. Using a series of tests and idealized models, we show that Taichi can accurately model collisional effects, such as dynamical friction and the core-collapse time of idealized clusters, producing results in strong agreement with benchmarks from other collisional codes such as NBODY6++GPU or PeTar. Parallelized using OpenMP and AVX, Taichi is demonstrated to be more efficient than other CPU-based direct N-body codes for simulating large systems. With future improvements to the handling of close encounters and binary evolution, we clearly demonstrate the potential of an optimized FMM for the modeling of collisional stellar systems, opening the door to accurate simulations of massive globular clusters, super star clusters, and even galactic nuclei.

Do star clusters form in a completely mass-segregated way?

Context. ALMA observations of the Serpens South star-forming region suggest that stellar protoclusters may be completely mass segregated at birth. Independent observations also suggest that embedded clusters form segregated by mass. Aims. As the primordial mass segregation seems to be lost over time, we aim to study on which timescale an initially perfectly mass-segregated star cluster becomes indistinguishable from an initially not mass-segregated cluster. As an example, the Orion Nebula Cluster (ONC) is also discussed. Methods. We used N-body simulations of star clusters with various masses and two different degrees of primordial mass segregation. We analysed their energy redistribution through two-body relaxation to quantify the time when the models agree in terms of mass segregation, which sets in only dynamically in the models that are primordially not mass segregated. A comprehensive cross-matched catalogue combining optical, infrared, and X-ray surveys of ONC members was also compiled and made available. Results. The models evolve to a similar radial distribution of high-mass stars after the core collapse (about half a median two-body relaxation time, trh) and become observationally indistinguishable from the point of view of mass segregation at time τv ≈ 3.3 trh. In the case of the ONC, using the distribution of high-mass stars, we may not rule out either evolutionary scenario (regardless of whether they are initially mass segregated). When we account for extinction and elongation of the ONC, as reported elsewhere, an initially perfectly mass-segregated state seems to be more consistent with the observed cluster.

Merging black holes in young star clusters

Searching for distinctive signatures, which characterize different formation channels of binary black holes (BBHs), is a crucial step towards the interpretation of current and future gravitational wave detections. Here, we investigate the demography of merging BBHs in young star clusters (SCs), which are the nursery of massive stars. We performed $4\times{} 10^3$ N-body simulations of SCs with metallicity $Z=0.002$, initial binary fraction $0.4$ and fractal initial conditions, to mimic the clumpiness of star forming regions. Our simulations include a novel population-synthesis approach based on the code MOBSE. We find that SC dynamics does not affect the merger rate significantly, but leaves a strong fingerprint on the properties of merging BBHs. More than 50 % of merging BBHs in young SCs form by dynamical exchanges in the first few Myr. Dynamically formed merging BBHs are significantly heavier than merging BBHs in isolated binaries: merging BBHs with total mass up to $\sim{}120$ M$_\odot$ form in young SCs, while the maximum total mass of merging BBHs in isolated binaries with the same metallicity is only $\sim{}70$ M$_\odot$. Merging BBHs born via dynamical exchanges tend to have smaller mass ratios than BBHs in isolated binaries. Furthermore, SC dynamics speeds up the merger: the delay time between star formation and coalescence is significantly shorter in young SCs. In our simulations, massive systems such as GW170729 form only via dynamical exchanges. Finally $\sim{}2$ % of merging BBHs in young SCs have mass in the pair-instability mass gap ($\sim{}60-120$ M$_\odot$). This represents a unique fingerprint of merging BBHs in SCs.

The distribution of stars around the Milky Way’s central black hole

Context. The distribution of stars around a massive black hole (MBH) has been addressed in stellar dynamics for the last four decades by a number of authors. Because of its proximity, the centre of the Milky Way is the only observational test case where the stellar distribution can be accurately tested. Past observational work indicated that the brightest giants in the Galactic centre (GC) may show a density deficit around the central black hole, not a cusp-like distribution, while we theoretically expect the presence of a stellar cusp. Aims. We here present a solution to this long-standing problem. Methods. We performed direct-summation N-body simulations of star clusters around massive black holes and compared the results of our simulations with new observational data of the GC’s nuclear cluster. Results. We find that after a Hubble time, the distribution of bright stars as well as the diffuse light follow power-law distributions in projection with slopes of Γ ≈ 0.3 in our simulations. This is in excellent agreement with what is seen in star counts and in the distribution of the diffuse stellar light extracted from adaptive-optics (AO) assisted near-infrared observations of the GC. Conclusions. Our simulations also confirm that there exists a missing giant star population within a projected radius of a few arcsec around Sgr A*. Such a depletion of giant stars in the innermost 0.1 pc could be explained by a previously present gaseous disc and collisions, which means that a stellar cusp would also be present at the innermost radii, but in the form of degenerate compact cores.

The global mass functions of 35 Galactic globular clusters – II. Clues on the initial mass function and black hole retention fraction

In this paper we compare the mass function slopes of Galactic globular clusters recently determined by Sollima & Baumgardt (2017) with a set of dedicated N-body simulations of star clusters containing between 65,000 to 200,000 stars. We study clusters starting with a range of initial mass functions (IMFs), black hole retention fractions and orbital parameters in the parent galaxy. We find that the present-day mass functions of globular clusters agree well with those expected for star clusters starting with Kroupa or Chabrier IMFs, and are incompatible with clusters starting with single power-law mass functions for the low-mass stars. The amount of mass segregation seen in the globular clusters studied by Sollima & Baumgardt (2017) can be fully explained by two-body relaxation driven mass segregation from initially unsegregated star clusters. Based on the present-day global mass functions, we expect that a typical globular cluster in our sample has lost about 75% of its mass since formation, while the most evolved clusters have already lost more than 90% of their initial mass and should dissolve within the next 1 to 2 Gyr. Most clusters studied by Sollima & Baumgardt also show a large difference between their central and global MF slopes, implying that the majority of Galactic globular clusters is either near or already past core collapse. The strong mass segregation seen in most clusters also implies that only a small fraction of all black holes formed in globular clusters still reside in them.

The early evolution of star clusters in compressive and extensive tidal fields

Abstract We present N-body simulations of star clusters that initially evolve within a strong compressive tidal field and then transition into an extensive tidal fields of varying strengths. While subject to compressive tides, clusters can undergo significant heating due to two-body interactions and mass-loss due to the stellar evolution. When the cluster transitions into an extensive tidal field it is supervirialized, which leads to a rapid expansion and significant mass-loss before the cluster reaches virial equilibrium. After the transition, clusters are significantly less massive, more extended and therefore more tidally filling than clusters that have spent their entire lifetimes in a similar extensive tidal field.

N-body modelling of globular clusters: masses, mass-to-light ratios and intermediate-mass black holes

We have determined the masses and mass-to-light ratios of 50 Galactic globular clusters by comparing their velocity dispersion and surface brightness profiles against a large grid of 900 N-body simulations of star clusters of varying initial concentration, size and central black hole mass fraction. Our models follow the evolution of the clusters under the combined effects of stellar evolution and two-body relaxation allowing us to take the effects of mass segregation and energy equipartition between stars self-consistently into account. For a subset of 16 well observed clusters we also derive their kinematic distances. We find an average mass-to-light ratio of Galactic globular clusters of $<M/L_V>=1.98 \pm 0.03$, which agrees very well with the expected M/L ratio if the initial mass function of the clusters was a standard Kroupa or Chabrier mass function. We do not find evidence for a decrease of the average mass-to-light ratio with metallicity. The surface brightness and velocity dispersion profiles of most globular clusters are incompatible with the presence of intermediate-mass black holes (IMBHs) with more than a few thousand $M_\odot$ in them. The only clear exception is $\omega$ Cen, where the velocity dispersion profile provides strong evidence for the presence of a $\sim$40,000 $M_\odot$ IMBH in the centre of the cluster.

Merged or monolithic? Using machine-learning to reconstruct the dynamical history of simulated star clusters

Context. Machine-Learning (ML) solves problems by learning patterns from data, with limited or no human guidance. In Astronomy, it is mainly applied to large observational datasets, e.g. for morphological galaxy classification. Aims. We apply ML to gravitational N-body simulations of star clusters that are either formed by merging two progenitors or evolved in isolation, planning to later identify Globular Clusters (GCs) that may have a history of merging from observational data. Methods. We create mock-observations from simulated GCs, from which we measure a set of parameters (also called features in the machine-learning field). After dimensionality reduction on the feature space, the resulting datapoints are fed to various classification algorithms. Using repeated random subsampling validation we check whether the groups identified by the algorithms correspond to the underlying physical distinction between mergers and monolithically evolved simulations. Results. The three algorithms we considered (C5.0 trees, k-nearest neighbour, and support-vector machines) all achieve a test misclassification rate of about 10% without parameter tuning, with support-vector machines slightly outperforming the others. The first principal component of feature space correlates with cluster concentration. If we exclude it from the regression, the performance of the algorithms is only slightly reduced.

Mass segregation in star clusters is not energy equipartition

Abstract Mass segregation in star clusters is often thought to indicate the onset of energy equipartition, where the most massive stars impart kinetic energy to the lower-mass stars and brown dwarfs/free-floating planets. The predicted net result of this is that the centrally concentrated massive stars should have significantly lower velocities than fast-moving low-mass objects on the periphery of the cluster. We search for energy equipartition in initially spatially and kinematically substructured N-body simulations of star clusters with N = 1500 stars, evolved for 100 Myr. In clusters that show significant mass segregation we find no differences in the proper motions or radial velocities as a function of mass. The kinetic energies of all stars decrease as the clusters relax, but the kinetic energies of the most massive stars do not decrease faster than those of lower-mass stars. These results suggest that dynamical mass segregation – which is observed in many star clusters – is not a signature of energy equipartition from two-body relaxation.

Biases in the determination of dynamical parameters of star clusters: today and in the Gaia era

The structural and dynamical properties of star clusters are generally derived by means of the comparison between steady-state analytic models and the available observables. With the aim of studying the biases of this approach, we fitted different analytic models to simulated observations obtained from a suite of direct N-body simulations of star clusters in different stages of their evolution and under different levels of tidal stress to derive mass, mass function and degree of anisotropy. We find that masses can be under/over-estimated up to 50% depending on the degree of relaxation reached by the cluster, the available range of observed masses and distances of radial velocity measures from the cluster center and the strength of the tidal field. The mass function slope appears to be better constrainable and less sensitive to model inadequacies unless strongly dynamically evolved clusters and a non-optimal location of the measured luminosity function are considered. The degree and the characteristics of the anisotropy developed in the N-body simulations are not adequately reproduced by popular analytic models and can be detected only if accurate proper motions are available. We show how to reduce the uncertainties in the mass, mass-function and anisotropy estimation and provide predictions for the improvements expected when Gaia proper motions will be available in the near future.

Reconstructing the initial mass function of disc–bulge Galactic globular clusters from N-body simulations

We propose an evolutionary model to describe the dynamical evolution of star cluster systems in tidal fields, in which we calibrated the parametric equations defining the model by running direct N-body simulations of star clusters with a wide range of initial masses and set of orbital parameters, living within the external tidal field generated by a disc-like galaxy. We derived a new method to solve numerically the evolutionary equations, allowing us to infer constraints on the mass of a star cluster from its age, present-day mass, orbital parameters and external gravitational potential. The result has been applied to the metal-rich subsample of Galactic globular clusters, being a good representation of a disc-bulge population. We reconstructed the initial mass function of these objects from the present-day mass function, finding that a lognormal distribution is well preserved during the evolution of the globular cluster system. The evolution of a power-law initial mass function has been evaluated, confirming that it transforms into a lognormal distribution of the cluster masses within an Hubble time. Our results are consistent with a formation scenario in which metal-rich Galactic globular clusters formed from giant molecular clouds in high-pressure regions during the early phases of the evolution of the Galactic disc and bulge.

The role of galaxy mergers on the evolution of star clusters

Abstract Interacting galaxies favour the formation of star clusters but are also suspected to affect their evolution through an intense and rapidly varying tidal field. Treating this complex behaviour remains out-of-reach of (semi-)analytical studies. By computing the tidal field from galactic models and including it into star-by-star N-body simulations of star clusters, we monitor the structure and mass evolution of a population of clusters in a galaxy major merger, taking the Antennae galaxies (NGC 4038/39) as a prototype. On the long time-scale (∼109 yr), the merger only indirectly affects the evolution of clusters by modifying their orbits in or around the galaxies: the mass-loss of clusters in the merger remnant is faster, while clusters ejected in the tidal debris survive much longer, compared to in an isolated galaxy. The tidal perturbations of the galactic collisions themselves are too short lived and not strong enough to significantly influence the structure and dissolution of realistically dense/massive star clusters.

THE INFLUENCE OF ORBITAL ECCENTRICITY ON TIDAL RADII OF STAR CLUSTERS

We have performed N-body simulations of star clusters orbiting in a spherically symmetric smooth galactic potential. The model clusters cover a range of initial half-mass radii and orbital eccentricities in order to test the historical assumption that the tidal radius of a cluster is imposed at perigalacticon. The traditional assumption for globular clusters is that since the internal relaxation time is larger than its orbital period, the cluster is tidally stripped at perigalacticon. Instead, our simulations show that a cluster with an eccentric orbit does not need to fully relax in order to expand. After a perigalactic pass, a cluster re-captures previously unbound stars, and the tidal shock at perigalacticon has the effect of energizing inner region stars to larger orbits. Therefore, instead of the limiting radius being imposed at perigalacticon, it more nearly traces the instantaneous tidal radius of the cluster at any point in the orbit. We present a numerical correction factor to theoretical tidal radii calculated at perigalacticon which takes into consideration both the orbital eccentricity and current orbital phase of the cluster.

Destruction of star clusters due to the radial migration in spiral galaxies

Abstract Most stars in galactic discs are believed to be born as a member of star clusters or associations. Star clusters formed in discs are disrupted due to the tidal stripping and the evolution of star clusters themselves, and as a result new stars are supplied to the galactic discs. We performed N-body simulations of star clusters in galactic discs, in which both star clusters and galactic discs are modelled as N-body (‘live’) systems, and as a consequence the discs form transient and recurrent spiral arms. In such non- steady spiral arms, star clusters migrate radially due to the interaction with spiral arms. We found that the migration time-scale is a few hundred Myr and that the angular momentum changes of star clusters are at most ∼50 per cent in 1 Gyr. Radial migration of star clusters to the inner region of galaxies results in a fast disruption of the star clusters because of a stronger tidal field in the inner region of the galaxy. This effect is not negligible for the disruption time-scale of star clusters in galactic discs. Stars stripped from clusters form tidal tails which spread over 1–2 kpc. While the spatial distribution of tidal tails changes in a complicated way due to the non-steady spiral arms, the velocity distribution conserves well even if the tidal tails are located at a few kpc from their parent clusters. Tidal tails of clusters in galactic discs might be detected using velocity plots.

THE SIZE SCALE OF STAR CLUSTERS

Direct N-body simulations of star clusters in a realistic Milky Way-like potential are carried out using the code NBODY6. Based on these simulations a new relationship between scale size and galactocentric distance is derived: the scale size of star clusters is proportional to the hyperbolic tangent of the galactocentric distance. The half-mass radius of star clusters increases systematically with galactocentric distance but levels off when star clusters orbit the galaxy beyond ~40 kpc. These simulations show that the half-mass radius of individual star clusters varies significantly as they evolve over a Hubble time, more so for clusters with shorter relaxation times, and remains constant through several relaxation times only in certain situations when expansion driven by the internal dynamics of the star cluster and the influence of the host galaxy tidal field balance each other. Indeed, the radius of a star cluster evolving within the inner 20 kpc of a realistic galactic gravitational potential is severely truncated by tidal interactions and does not remain constant over a Hubble time. Furthermore, the half-mass radius of star clusters measured with present day observations bears no memory of the original cluster size. Stellar evolution and tidal stripping are the two competing physical mechanisms that determine the present day size of globular clusters. These simulations also show that extended star clusters can form at large galactocentric distances while remaining fully bound to the host galaxy. There is thus no need to invoke accretion from an external galaxy to explain the presence of extended clusters at large galactocentric distances in a Milky Way-type galaxy.

Star cluster evolution in barred disc galaxies - I. Planar periodic orbits

The dynamical evolution of stellar clusters is driven to a large extent by their environment. Several studies so far have considered the effect of tidal fields and their variations, such as, e.g., from giant molecular clouds, galactic discs, or spiral arms. In this paper we will concentrate on a tidal field whose effects on star clusters have not yet been studied, namely that of bars. We present a set of direct N-body simulations of star clusters moving in an analytic potential representing a barred galaxy. We compare the evolution of the clusters moving both on different planar periodic orbits in the barred potential and on circular orbits in a potential obtained by axisymmetrising its mass distribution. We show that both the shape of the underlying orbit and its stability have strong impact on the cluster evolution as well as the morphology and orientation of the tidal tails and the sub-structures therein. We find that the dissolution time-scale of the cluster in our simulations is mainly determined by the tidal forcing along the orbit and, for a given tidal forcing, only very little by the exact shape of the gravitational potential in which the cluster is moving.