The formation of a bound star cluster: from the Orion nebula cluster to the Pleiades

Direct N -body calculations are presented of the formation of Galactic clusters using GasEx , which is a variant of the code Nbody6 . The calculations focus on the possible evolution of the Orion nebula cluster (ONC) by assuming that the embedded OB stars explosively drove out 2/3 of its mass in the form of gas about 0.4 Myr ago. A bound cluster forms readily and survives for 150 Myr despite additional mass loss from the large number of massive stars, and the Galactic tidal field. This is the very first time that cluster formation is obtained under such realistic conditions. The cluster contains about 1/3 of the initial 10⁴ stars, and resembles the Pleiades cluster to a remarkable degree, implying that an ONC-like cluster may have been a precursor of the Pleiades. This scenario predicts the present expansion velocity of the ONC, which will be measurable by upcoming astrometric space missions. These missions should also detect the original Pleiades members as an associated expanding young Galactic-field subpopulation. The results arrived at here suggest that Galactic clusters form as the nuclei of expanding OB associations.
The results have wide implications, also for the formation of globular clusters and the Galactic-field and halo stellar populations. In view of this, the distribution of binary orbital periods and the mass function within and outside the model ONC and Pleiades is quantified, finding consistency with observational constraints. Advanced mass segregation is evident in one of the ONC models. The calculations show that the primordial binary population of both clusters could have been much the same as is observed in the Taurus–Auriga star-forming region. The computations also demonstrate that the binary proportion of brown dwarfs is depleted significantly for all periods, whereas massive stars attain a high binary fraction.     

On the infant weight loss of low- to intermediate-mass star clusters

Star clusters are born in a highly compact configuration, typically with radii of less than about 1 pc roughly independently of mass. Since the star formation efficiency is less than 50 per cent by observation and because the residual gas is removed from the embedded cluster, the cluster must expand. In the process of doing so it only retains a fraction f _st of its stars. To date there are no observational constraints for f _st, although N -body calculations by Kroupa, Aarseth & Hurley suggest it to be about 20–30 per cent for Orion-type clusters. Here we use the data compiled by Testi et al., Testi, Palla & Natta and Testi, Palla & Natta for clusters around young Ae/Be stars and by de Wit et al. and de Wit et al. around young O stars and the study of de Zeeuw et al. of OB associations and combine these measurements with the expected number of stars in clusters with primary Ae/Be and O stars, respectively, using the empirical correlation between maximal stellar mass and star cluster mass of Weidner & Kroupa. We find that   f _st < 50  per cent with a decrease to higher cluster masses/more massive primaries. The interpretation would be that cluster formation is very disruptive. It appears that clusters with a birth stellar mass in the range  10–10³ M_⊙  keep at most 50 per cent of their stars.     

Quantitative analysis of clumps in the tidal tails of star clusters

Tidal tails of star clusters are not homogeneous but show well-defined clumps in observations as well as in numerical simulations. Recently, an epicyclic theory for the formation of these clumps was presented. A quantitative analysis was still missing. We present a quantitative derivation of the angular momentum and energy distribution of escaping stars from a star cluster in the tidal field of the Milky Way and derive the connection to the position and width of the clumps. For the numerical realization we use star-by-star N -body simulations. We find a very good agreement of theory and models. We show that the radial offset of the tidal arms scales with the tidal radius, which is a function of cluster mass and the rotation curve at the cluster orbit. The mean radial offset is 2.77 times the tidal radius in the outer disc. Near the Galactic Centre the circumstances are more complicated, but to lowest order the theory still applies. We have also measured the Jacobi energy distribution of bound stars and showed that there is a large fraction of stars (about 35 per cent) above the critical Jacobi energy at all times, which can potentially leave the cluster. This is a hint that the mass loss is dominated by a self-regulating process of increasing Jacobi energy due to the weakening of the potential well of the star cluster, which is induced by the mass loss itself.     

Single star scattering on an open cluster

We examine the scattering of single stars from an open star cluster. The probability of the capture of a star by a star cluster is dependent on the velocity and mass of the star, and the stars that are not captured experience a velocity change. For low-velocity stars there is an exponential decrease of the capture probability with the initial velocity, and the velocity change decreases almost linearly. For high-velocity stars there is a v ⁻⁶ dependence for the capture probability, and a v ⁻¹ dependence for the velocity change. Analytical estimations, Monte Carlo and full N -body simulations are all in good agreement.     

High-velocity stars from decay of small stellar systems

In this paper we present numerical results on the decay of small stellar systems under different initial conditions (multiplicity 3 ≤  N  ≤ 10, and various mass spectra, initial velocities and initial configurations). The numerical treatment uses the CHAIN1 code (Mikkola &38; Aarseth). Particular attention is paid to the distribution of high-velocity escapers: we define these as stars with velocity above 30 km s⁻¹. These numerical experiments confirm that small N -body systems are dynamically unstable and produce cascades of escapers in the process of their decay. It is shown that the fraction of stars that escape from small dense stellar systems with an escape velocity greater than 30 km s⁻¹ is ∼1 per cent for all systems treated here. This relatively small fraction must be considered in relation to the rate of star formation in the Galaxy in small groups: this could explain some moderately high-velocity stars observed in the Galactic disc and possibly some young stars with relatively high metallicity in the thick disc.     

Influence of a stellar cusp on the dynamics of young stellar discs and the origin of the S-stars in the Galactic Centre

Observations of the Galactic Centre show evidence of one or two disc-like structures of very young stars orbiting the central supermassive black hole within a distance of a few 0.1 pc. A number of analyses have been carried out to investigate the dynamical behaviour and consequences of these discs, including disc thickness and eccentricity growth as well as mutual interaction and warping. However, most of these studies have neglected the influence of the stellar cusp surrounding the black hole, which is believed to be one to two orders of magnitude more massive than the disc(s).
By means of N -body integrations using our bhint code, we study the impact of stellar cusps of different compositions. We find that although the presence of a cusp does have an important effect on the evolution of an otherwise isolated flat disc, its influence on the evolution of disc thickness and warping is rather mild in a two-disc configuration. However, we show that the creation of highly eccentric orbits strongly depends on the graininess of the cusp (i.e. the mean and maximum stellar masses). While Chang recently found that full cycles of Kozai resonance are prevented by the presence of an analytic cusp, we show that relaxation processes play an important role in such highly dense regions and support short-term resonances. We thus find that young disc stars on initially circular orbits can achieve high eccentricities by resonant effects also in the presence of a cusp of stellar remnants, yielding a mechanism to create S-stars and hypervelocity stars.
Furthermore, we discuss the underlying initial mass function (IMF) of the young stellar discs and find no definite evidence for a non-canonical IMF.     

The influence of multiple stars on the high-mass stellar initial mass function and age dating of young massive star clusters

The study of young stellar populations has revealed that most stars are in binary or higher order multiple systems. In this study, the influence on the stellar initial mass function (IMF) of large quantities of unresolved multiple massive stars is investigated by taking into account the stellar evolution and photometrically determined system masses. The models, where initial masses are derived from the luminosity and colour of unresolved multiple systems, show that even under extreme circumstances (100 per cent binaries or higher order multiples), the difference between the power-law index of the mass function (MF) of all stars and the observed MF is small (≲0.1). Thus, if the observed IMF has the Salpeter index  α= 2.35  , then the true stellar IMF has an index not flatter than  α= 2.25  . Additionally, unresolved multiple systems may hide between 15 and 60 per cent of the underlying true mass of a star cluster. While already a known result, it is important to point out that the presence of a large number of unresolved binaries amongst pre-main-sequence stars induces a significant spread in the measured ages of these stars even if there is none. Also, lower mass stars in a single-age binary-rich cluster appear older than the massive stars by about 0.6 Myr.     

Monte Carlo simulations of star clusters – III. A million-body star cluster

A revision of Stodółkiewicz's Monte Carlo code is used to simulate the evolution of million-body star clusters. The new method treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. The evolution of N -body systems influenced by the tidal field of a parent galaxy and by stellar evolution is presented. All models consist of 1 000 000 stars. The process of energy generation is realized by means of appropriately modified versions of Spitzer's and Mikkola's formulae for the interaction cross-section between binaries and field stars and binaries themselves. The results presented are in good agreement with theoretical expectations and the results of other methods. During the evolution, the initial mass function (IMF) changes significantly. The local mass function around the half-mass radius closely resembles the actual global mass function. At the late stages of evolution, the mass of the evolved stars inside the core can be as high as 97 per cent of the total mass in this region. For the whole system, the evolved stars can compose up to 75 per cent of the total mass. The evolution of cluster anisotropy strongly depends on initial cluster concentration, IMF and the strength of the tidal field. The results presented are the first step in the direction of simulating the evolution of real globular clusters by means of the Monte Carlo method.     

Primordial substructure in the Orion Nebula Cluster

We use numerical N -body simulations of the Orion Nebula Cluster (ONC) to investigate the possibility of substructure in its formation. There is no substructure apparent in the ONC today. However, unless there was a remarkable degree of homogeneity in the molecular cloud from which it formed, it seems unlikely that this would have been true of the cluster in its earliest phase. More plausibly, the early structure of the cluster would have consisted of groups or clumps of subclusters, following the structure of the cloud itself. We have explored the extent to which such subclusters could subsequently have merged, and find that the age of the cluster is a critical factor. The most inhomogeneous initial conditions, comprising a small number of subclusters with many members, are ruled out by an age of 2 Myr or less. There is a considerable amount of freedom in the other direction, however, which suggests that fragmentation in the original cloud is more likely to have been on the scale of small clumps, each producing fewer than 100 stars. These initial subclusters could have been very dense – perhaps two or three orders of magnitude more dense than the core of the ONC today.     

Determination of the IMF on the basis of a recently derived SFR history in the solar neighbourhood

The initial mass function (IMF) in the solar neighbourhood is determined on the basis of a recently derived history of the star formation rate (SFR) which shows the presence of a star formation burst about 8 Gyr ago. The observed present-day mass function (PDMF) is considered, and the average past distribution of stars of a given mass is estimated. Two cases are considered, namely (i) constant SFR, and (ii) variable SFR as derived from the new metallicity distribution of G dwarfs. The resulting IMF is compared with previous determinations by Scalo and Kroupa et al., and the variation with stellar mass of the slope of the IMF is compared with reference determinations in the literature.     

Cosmological origin of the lowest metallicity halo stars

We explore the predictions of the standard hierarchical clustering scenario of galaxy formation, regarding the numbers and metallicities of PopIII stars that are likely to be found within our Galaxy today. By PopIII we refer to stars formed at large redshift ( z >4), with low metallicities ([ Z /Z_⊙]<−2.5) and in small systems (total mass ≲ 2×10⁸ M_⊙) that are extremely sensitive to stellar feedback, and which through a prescribed merging history end up becoming part of the Milky Way today. An analytic, extended Press–Schechter formalism is used to obtain the mass functions of haloes which will host PopIII stars at a given redshift, and which will end up in Milky Way sized systems today. Each of these is modelled as a mini-galaxy, with a detailed treatment of the dark halo structure, angular momentum distribution, final gas temperature and disc instabilities, all of which determine the fraction of the baryons that are subject to star formation. The use of new primordial metallicity stellar evolutionary models allows us to trace the history of the stars formed, and give accurate estimates of their expected numbers today and their location in L /L_⊙ versus T /K Hertzsprung–Russell (HR) diagrams. A first comparison with observational data suggests that the initial mass function (IMF) of the first stars was increasingly high-mass weighted towards high redshifts, levelling off at z ≳9 at a characteristic stellar mass scale m _s=10–15 M_⊙.     

Using the minimum spanning tree to trace mass segregation

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 (MSR)'  Λ_MSR= 8.0 ± 3.5  (where  Λ_MSR= 1  is no mass segregation) down to  16 M_⊙  , and also that the ONC is mass segregated at a lower level  (∼2.0 ± 0.5)  down to  5 M_⊙  . Below  5 M_⊙  we find no evidence for any further mass segregation in the ONC.     

The origin of the initial mass function and its dependence on the mean Jeans mass in molecular clouds

We investigate the dependence of stellar properties on the mean thermal Jeans mass in molecular clouds. We compare the results from the two largest hydrodynamical simulations of star formation to resolve the fragmentation process down to the opacity limit, the first of which was reported by Bate, Bonnell & Bromm. The initial conditions of the two calculations are identical except for the radii of the clouds, which are chosen so that the mean densities and mean thermal Jeans masses of the clouds differ by factors of 9 and 3, respectively.
We find that the denser cloud, with the lower mean thermal Jeans mass, produces a higher proportion of brown dwarfs and has a lower characteristic (median) mass of the stars and brown dwarfs. This dependence of the initial mass function (IMF) on the density of the cloud may explain the observation that the Taurus star-forming region appears to be deficient in brown dwarfs when compared with the Orion Trapezium cluster. The new calculation also produces wide binaries (separations >20 au), one of which is a wide binary brown dwarf system.
Based on the hydrodynamical calculations, we develop a simple accretion/ejection model for the origin of the IMF. In the model, all stars and brown dwarfs begin with the same mass (set by the opacity limit for fragmentation) and grow in mass until their accretion is terminated stochastically by their ejection from the cloud through dynamically interactions. The model predicts that the main variation of the IMF in different star-forming environments should be in the location of the peak (due to variations in the mean thermal Jeans mass of the cloud) and in the substellar regime. However, the slope of the IMF at high masses may depend on the dispersion in the accretion rates of protostars.     

Constraints on stellar-dynamical models of the Orion Nebula Cluster

The results obtained by Kroupa, Petr & McCaughrean (1999) for specific models of young compact binary-rich clusters are generalised using dynamical scaling relations, to infer the candidate set of possible birth models leading to the Orion Nebula Cluster (ONC), of which the Trapezium Cluster is the core. It is found that candidate sets of solutions exist which allow the ONC to be in virial equilibrium, expanding or contracting. The range of possible solutions is quite narrow. These results will serve as guidelines for future, CPU-intensive calculations of the stellar-dynamical and astrophysical evolution of the entire ONC. These, in turn, will be essential to quantify observables that will ultimately discriminate between models, thus allowing us to understand if the ONC is in the process of assembling a rich Galactic cluster, and, if this is the case, how it occurs.     

Mass segregation in young stellar clusters

We investigate the evolutionary effect of dynamical mass segregation in young stellar clusters. Dynamical mass segregation acts on a time-scale of order the relaxation time of a cluster. Although some degree of mass segregation occurs earlier, the position of massive stars in rich young clusters generally reflects the cluster's initial conditions. In particular, the positions of the massive stars in the Trapezium cluster in Orion cannot be due to dynamical mass segregation, but indicate that they formed in, or near, the centre of the cluster. Implications of this for cluster formation and for the formation of high-mass stars are discussed.     

Binary stars and the fundamental initial mass function

We have carried out Monte Carlo simulations in which we generate a random pairing of objects drawn from a pre-assumed single-star power-law initial mass function (IMF), which we call the fundamental IMF. We show how the mass functions of primary stars and secondary stars and the mass function of the total mass of systems (if we could resolve them) differ from the underlying fundamental IMF for different slopes of this IMF. We also compare our results with the observed IMF, the binary frequency and the binary mass-ratio distributions for field stars and conclude that the fundamental IMF of subsolar mass stars could be steeper than is currently believed. In other words, the low-mass turn-over of the observed ('apparent') IMF could be spurious, if the main-sequence binary fraction of field stars is close to 100 per cent (perhaps owing to invisible companions).     

OB星协和年轻星团的形成与早期演化

ＯＢ星协和年轻星团是恒星形成与早期演化的“化石”，同时也是研究初始重质量函数（ＩＭＦ）的最好场所；文中就ＯＢ星协和年轻星团的形成和早期演化方面的研究进展作了一评述，还论述了ＩＭＦ的测定和研究情况，并对相关的速逃ＯＢ星及蓝离散星问题作了简要介绍。     

A search for previously unrecognized metal-poor subdwarfs in the Hipparcos astrometric catalogue

We have identified 317 stars included in the Hipparcos astrometric catalogue that have parallaxes measured to a precision of better than 15 per cent, and the location of which in the ( M _V ,( B − V ) _T ) diagram implies a metallicity comparable to or less than that of the intermediate-abundance globular cluster M5. We have undertaken an extensive literature search to locate Strömgren, Johnson/Cousins and Walraven photometry for over 120 stars. In addition, we present new UBV ( RI )_C photometry of 201 of these candidate halo stars, together with similar data for a further 14 known metal-poor subdwarfs. These observations provide the first extensive data set of R _C I _C photometry of metal-poor, main-sequence stars with well-determined trigonometric parallaxes. Finally, we have obtained intermediate-resolution optical spectroscopy of 175 stars.
47 stars still lack sufficient supplementary observations for population classification; however, we are able to estimate abundances for 270 stars, or over 80 per cent of the sample. The overwhelming majority have near-solar abundance, with their inclusion in the present sample stemming from errors in the colours listed in the Hipparcos catalogue. Only 44 stars show consistent evidence of abundances below [Fe/H]=−1.0 . Nine are additions to the small sample of metal-poor subdwarfs with accurate photometry. We consider briefly the implication of these results for cluster main-sequence fitting.     

High resolution study of the abundance pattern of the heavy elements in very metal‐poor field stars

The abundances of heavy elements in EMP stars are not well explained by the simple view of an initial basic “rapid” process. In a careful and homogeneous analysis of the “First Stars” sample (eighty per cent of the stars have a metallicity [Fe/H] ≃ –3.1 ± 0.4), it has been shown that at this metallicity [Eu/Ba] is constant, and therefore the europium‐rich stars (generally called “r‐rich”) are also Ba‐rich. The very large variation of [Ba/Fe] (existence of “r‐poor” and “r‐rich” stars) induces that the early matter was not perfectly mixed. On the other hand, the distribution of the values of [Sr/Ba] vs. [Ba/Fe] appears with well defined upper and lower envelopes. No star was found with [Sr/Ba] < –0.5 and the scatter of [Sr/Ba] increases regularly when [Ba/Fe] decreases. To explain this behavior, we suggest that an early “additional” process forming mainly first peak elements would affect the initial composition of the matter. For a same quantity of accreted matter, this additional Sr production would barely affect the r‐rich matter (which already contains an important quantity of Sr) but would change significantly the composition of the r‐poor matter. The abundances found in the CEMP‐r+s stars reflect the transfer of heavy elements from a defunct AGB companion. But the abundances of the heavy elements in CEMP‐no stars present the same characteristics as the the abundances in the EMP stars. Direct stellar ages may be found from radioactive elements, the precision is limited by the precision in the measurements of abundances from faint lines in faint stars, and the uncertainty in the initial abundances of the radioactive elements. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)