Search

We present the results of a photometric search for variable stars in the core of the Galactic globular cluster M4. The input data are a large and unprecedented set of deep Hubble Space Telescope WFC3 images (large program GO-12911; 120 orbits allocated), primarily aimed at probing binaries with massive companions by detecting their astrometric wobbles. Though these data were not optimised to carry out a time-resolved photometric survey, their exquisite precision, spatial resolution and dynamic range enabled us to firmly detect 38 variable stars, of which 20 were previously unpublished. They include 19 cluster-member eclipsing binaries (confirming the large binary fraction of M4), RR Lyrae, and objects with known X-ray counterparts. We improved and revised the parameters of some among published variables., Comment: 11 pages, 5 figures, 2 tables. Accepted for publication in MNRAS

The M4 Core Project with HST is designed to exploit the Hubble Space Telescope to investigate the central regions of M4, the Globular Cluster closest to the Sun. In this paper we combine optical and near-infrared photometry to study multiple stellar populations in M4. We detected two sequences of M-dwarfs containing ~38% (MS_I) and ~62% (MS_II) of MS stars below the main-sequence (MS) knee. We compare our observations with those of NGC2808, which is the only other GCs where multiple MSs of very low-mass stars have been studied to date. We calculate synthetic spectra for M-dwarfs, assuming the chemical composition mixture inferred from spectroscopic studies of stellar populations along the red giant branch, and different Helium abundances, and we compare predicted and observed colors. Observations are consistent with two populations, one with primordial abundance and another with enhanced nitrogen and depleted oxygen., Comment: 8 pages, 6 figure, accepted for publication in MNRAS

We present an overview of the ongoing Hubble Space Telescope large program GO-12911. The program is focused on the core of M4, the nearest Galactic globular cluster, and the observations are designed to constrain the number of binaries with massive companions (black holes, neutron stars, or white dwarfs) by measuring the ``wobble'' of the luminous (main-sequence) companion around the center of mass of the pair, with an astrometric precision of ~50 micro-arcseconds. The high spatial resolution and stable medium-band PSFs of WFC3/UVIS will make these measurements possible. In this work we describe: (i) the motivation behind this study, (ii) our observing strategy, (iii) the many other investigations enabled by this unique data set, and which of those our team is conducting, and (iv) a preliminary reduction of the first-epoch data-set collected on October 10, 2012., Comment: 25 pages, 14 figures (9 at low resolution), 3 tables. Published in: Astronomische Nachrichten, Volume 334, Issue 10, pages 1062-1085, December 2013. http://onlinelibrary.wiley.com/doi/10.1002/asna.201311911/abstract

Based on our recent work on tidal tails of star clusters (Kuepper et al. 2009) we investigate star clusters of a few 10^4 Msun by means of velocity dispersion profiles and surface density profiles. We use a comprehensive set of $N$-body computations of star clusters on various orbits within a realistic tidal field to study the evolution of these profiles with time, and ongoing cluster dissolution From the velocity dispersion profiles we find that the population of potential escapers, i.e. energetically unbound stars inside the Jacobi radius, dominates clusters at radii above about 50% of the Jacobi radius. Beyond 70% of the Jacobi radius nearly all stars are energetically unbound. The velocity dispersion therefore significantly deviates from the predictions of simple equilibrium models in this regime. We furthermore argue that for this reason this part of a cluster cannot be used to detect a dark matter halo or deviations from Newtonian gravity. By fitting templates to the about 10^4 computed surface density profiles we estimate the accuracy which can be achieved in reconstructing the Jacobi radius of a cluster in this way. We find that the template of King (1962) works well for extended clusters on nearly circular orbits, but shows significant flaws in the case of eccentric cluster orbits. This we fix by extending this template with 3 more free parameters. Our template can reconstruct the tidal radius over all fitted ranges with an accuracy of about 10%, and is especially useful in the case of cluster data with a wide radial coverage and for clusters showing significant extra-tidal stellar populations. No other template that we have tried can yield comparable results over this range of cluster conditions. All templates fail to reconstruct tidal parameters of concentrated clusters, however. (abridged), Comment: 23 pages, 13 figures, accepted for publication in MNRAS

Based on recent findings of a formation mechanism of substructure in tidal tails by Kuepper, Macleod & Heggie (2008) we investigate a more comprehensive set of N-body models of star clusters on orbits about a Milky-Way-like potential. We find that the predicted epicyclic overdensities arise in any tidal tail no matter which orbit the cluster follows as long as the cluster lives long enough for the overdensities to build up. The distance of the overdensities along the tidal tail from the cluster centre depends for circular orbits only on the mass of the cluster and the strength of the tidal field, and therefore decreases monotonically with time, while for eccentric orbits the orbital motion influences the distance, causing a periodic compression and stretching of the tails and making the distance oscillate with time. We provide an approximation for estimating the distance of the overdensities in this case. We describe an additional type of overdensity which arises in extended tidal tails of clusters on eccentric orbits, when the acceleration of the tidal field on the stellar stream is no longer homogeneous. Moreover, we conclude that a pericentre passage or a disk shock is not the direct origin of an overdensity within a tidal tail. Escape due to such tidal perturbations does not take place immediately after the perturbation but is rather delayed and spread over the orbit of the cluster. All observable overdensities are therefore of the mentioned two types. In particular, we note that substructured tidal tails do not imply the existence of dark-matter sub-structures in the haloes of galaxies., Comment: 20 pages, 17 figures, MNRAS accepted

We describe Monte Carlo models for the dynamical evolution of the nearby globular cluster NGC 6397. The code includes treatments of two-body relaxation, most kinds of three- and four-body interactions involving primordial binaries and those formed dynamically, the Galactic tide, and the internal evolution of both single and binary stars. We arrive at a set of initial parameters for the cluster which, after 12Gyr of evolution, gives a model with a fairly satisfactory match to the surface brightness profile, the velocity dispersion profile, and the luminosity function in two fields. We describe in particular those aspects of the evolution which distinguish this cluster from M4, which has a roughly similar mass and Galactocentric distance, but a qualitatively different surface brightness profile. Within the limitations of our modelling, we conclude that the most plausible explanation for the difference is fluctuations: both clusters are post-collapse objects, but sometimes have resolvable cores and sometimes not., Comment: 11 pages, 12 figures, MNRAS, in press; revised in response to referee report

In this study we consider the relationship between the survival time and the Lyapunov time for 3-body systems. It is shown that the Sitnikov problem exhibits a two-part power law relationship as demonstrated previously for the general 3-body problem. Using an approximate Poincare map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the 3-body problem. The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general 3-body problem., Comment: 9 pages, 19 figures, accepted for publication in MNRAS

We present a large multi-epoch high resolution spectroscopic investigation for the search of binary candidates in the Galactic Globular Cluster (GGC) M4. The aim of our work is the identification of the binary candidates, and the determination of the binary fraction and of the binary radial distribution. We present a large multi-epoch high resolution spectroscopic investigation for the search of binary candidates in the Galactic Globular Cluster (GGC) M4. The aim of our work is the identification of the binary candidates, and the determination of the binary fraction and of the binary radial distribution. The average radial velocity of the observed cluster members is $70.29+/-0.07(+/-0.3)(+/-0.1)km/s. The search for variations in radial velocities among the stars with multi-epoch observations yielded 57 binary star candidates. Our radial velocity measurement accuracy allowed us to identify at a 3sigma level binaries with radial velocity variations larger than ~0.3km/s for the target stars with V<15, and larger than ~0.5km/s for the targets with V>15. We identified 4 binary star candidates out of 97 observed targets inside the core radius, and 53 candidates out of 2372 observed stars outside the core radius. Accounting for the incompleteness affecting our survey, the lower limit for the total binary fraction is f=3.0+/-0.3%. The lower limit for the binary fraction in the cluster core is f=5.1+/-2.3%, while outside the core it decreases to f=3.0+/-0.4%. Similarly, we found f=4.5+/-0.4% and f=1.8+/-0.6% for the binary fraction inside and outside the half mass radius., Comment: 13 pages, 12 figures, Accepted for publication in A&A

Astronomers have constructed models of globular clusters for over 100 years. These models mainly fall into two categories: (i) static models, such as King's model and its variants, and (ii) evolutionary models. Most attention has been given to static models, which are used to estimate mass-to-light ratios and mass segregation, and to combine data from proper motions and radial velocities. Evolutionary models have been developed for a few objects using the gaseous model, the Fokker-Planck model, Monte Carlo models and N-body models. These models have had a significant role in the search for massive black holes in globular clusters, for example. In this presentation the problems associated with these various techniques will be summarised, and then we shall describe new work with Giersz's Monte Carlo code, which has been enhanced recently to include the stellar evolution of single and binary stars. We describe in particular recent attempts to model the nearby globular cluster M4, including predictions on the spatial distribution of binary stars and their semi-major axis distribution, to illustrate the effects of about 12 Gyr of dynamical evolution. We also discuss work on an approximate way of predicting the "initial" conditions for such modelling., Comment: 10 pages,9 figures,invited review, to appear in the proceedings of IAU Symposium 246, 'Dynamical Evolution of Dense Stellar Systems', eds. E. Vesperini, M. Giersz, A. Sills, Capri, Sept. 2007. Sent to the editors 9 October 2007

Though about 80 pulsar binaries have been detected in globular clusters so far, no pulsar has been found in a triple system in which all three objects are of comparable mass. Here we present predictions for the abundance of such triple systems, and for the most likely characteristics of these systems. Our predictions are based on an extensive set of more than 500 direct simulations of star clusters with primordial binaries, and a number of additional runs containing primordial triples. Our simulations employ a number N_{tot} of equal mass stars from N_{tot}=512 to N_{tot}=19661 and a primordial binary fraction from 0-50%. In addition, we validate our results against simulations with N=19661 that include a mass spectrum with a turn-off mass at 0.8 M_{sun}, appropriate to describe the old stellar populations of galactic globular clusters. Based on our simulations, we expect that typical triple abundances in the core of a dense cluster are two orders of magnitude lower than the binary abundances, which in itself already suggests that we don't have to wait too long for the first comparable-mass triple with a pulsar to be detected., Comment: 11 pages, minor changes to match MNRAS accepted version

At least 10-15% of nearby sun-like stars have known Jupiter-mass planets. In contrast, very few planets are found in mature open and globular clusters such as the Hyades and 47 Tuc. We explore here the possibility that this dichotomy is due to the post-formation disruption of planetary systems associated with the stellar encounters in long-lived clusters. One supporting piece of evidence for this scenario is the discovery of freely floating low-mass objects in star forming regions. We use two independent numerical approaches, a hybrid Monte Carlo and a direct $N$-body method, to simulate the impact of the encounters. We show that the results of numerical simulations are in reasonable agreement with analytical determinations in the adiabatic and impulsive limits. They indicate that distant stellar encounters generally do not significantly modify the compact and nearly circular orbits. However, moderately close stellar encounters, which are likely to occur in dense clusters, can excite planets' orbital eccentricity and induce dynamical instability in systems which are closely packed with multiple planets. We compute effective cross sections for the dissolution of planetary systems and show that, for all initial eccentricities, dissolution occurs on time scales which are longer than the dispersion of small stellar associations, but shorter than the age of typical open and globular clusters. Although it is much more difficult to disrupt short-period planets, close encounters can excite modest eccentricity among them, such that subsequent tidal dissipation leads to orbital decay, tidal inflation, and even disruption of the close-in planets., Comment: 57 pages, 14 figures, 4 tables, major revision by authors, accepted for publication at the Astrophysical Journal (ApJ)

In order to interpret the results of complex realistic star cluster simulations, which rely on many simplifying approximations and assumptions, it is essential to study the behavior of even more idealized models, which can highlight the essential physical effects and are amenable to more exact methods. With this aim, we present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, with values of N ranging from 256 to 16384. This allows us to extrapolate the main features of the evolution to systems comparable in particle number with globular clusters. In this range, we find that the steady-state `deuterium main sequence' is characterized by a ratio of the core radius to half-mass radius that follows qualitatively the analytical estimate by Vesperini & Chernoff (1994), although the N dependence is steeper than expected. Interestingly, for an initial binary fraction f greater than 10%, the binary heating in the core during the post collapse phase almost saturates (becoming nearly independent of f), and so little variation in the structural properties is observed. Thus, although we observe a significantly lower binary abundance in the core with respect to the Fokker-Planck simulations by Gao et al. (1991), this is of little dynamical consequence. At variance with the study of Gao et al. (1991), we see no sign of gravothermal oscillations before 150 halfmass relaxation times. At later times, however, oscillations become prominent. We demonstrate the gravothermal nature of these oscillations., Comment: 14 pages, 22 figures, MNRAS accepted

[abridged] We extend our analysis of the dynamical evolution of simple star cluster models, in order to provide comparison standards that will aid in interpreting the results of more complex realistic simulations. We augment our previous primordial-binary simulations by introducing a tidal field, and starting with King models of different central concentrations. We present the results of N-body calculations of the evolution of equal-mass models, starting with primordial binary fractions of 0 - 100 %, and N values from 512 to 16384. We also attempt to extrapolate some of our results to the larger number of particles that are necessary to model globular clusters. We characterize the steady-state `deuterium main sequence' phase in which primordial binaries are depleted in the core in the process of `gravitationally burning'. In this phase we find that the ratio of the core to half-mass radius, r_c/r_h, is similar to that measured for isolated systems. In addition to the generation of energy due to hardening and depletion of the primordial binary population, the overall evolution of the star clusters is driven by a competing process: the tidal disruption of the system. We find that the depletion of primordial binaries before tidal dissolution of the system is possible only if the initial number is below 0.05 N, in the case of a King model with W_0=7 and N=4096 (which is one of our longest living models). We compare our findings, obtained by means of direct N-body simulations but scaled, where possible, to larger N, with similar studies carried out by means of Monte Carlo methods., Comment: 15 pages, 18 figures, matches MNRAS accepted version, some sections reorganized but no major changes

We point out a strong time-evolution of the mass-to-light conversion factor \eta commonly used to estimate masses of dense star clusters from observed cluster radii and stellar velocity dispersions. We use a gas-dynamical model coupled with the Cambridge stellar evolution tracks to compute line-of-sight velocity dispersions and half-light radii weighted by the luminosity. Stars at birth are assumed to follow the Salpeter mass function in the range [0.15--17 M_\sun]. We find that $\eta$, and hence the estimated cluster mass, increases by factors as large as 3 over time-scales of 20 million years. Increasing the upper mass limit to $50 M_\sun$ leads to a sharp rise of similar amplitude but in as little as 10 million years. Fitting truncated isothermal (Michie-King) models to the projected light profile leads to over-estimates of the concentration par ameter c of $\delta c\approx 0.3$ compared to the same functional fit applied to the proj ected mass density., Comment: Draft version of an ApJ letter

Star clusters and dwarf galaxies gradually dissolve as they move in the potential of their host galaxy. Once their density falls below a certain critical density (which is comparable with the background density of the galaxy) it is often assumed that their evolution is completed. In fact the remnant of such a system forms a distribution of stars which are unbound to each other and which move on similar orbits in their host potential. With this motivation we study the evolution of an idealised unbound system and follow its expansion and dissolution in the tidal field of a model galaxy. Initially the stars are uniformly distributed (with a density below the critical density) within an ellipsoidal volume. The system itself travels on a circular orbit within a galaxy modelled as an isothermal sphere. The initial velocities of the stars are chosen by assuming that they move on (three-dimensional) epicycles with guiding centre at the centre of the ellipsoid, though the usual epicyclic theory is altered to account for the self-gravity of the system. This is believed to be the first exact equilibrium model of a stellar system in a tidal field. Our main task is to study the stability of such configurations and the time-scale of their dissolution, as a function of the initial density and size of the ellipsoid. If the time of dissolution is measured by an increase of the half-mass radius of 50 %, we find that systems of low density (about 1 % of the background density) and small radius (50 pc on an orbit of radius 10 kpc) can survive for about 20 galactic orbits. For small systems we show that the lifetime is approximately proportional to the inverse square root of the density., Comment: 14 pages, 7 figures, accepted by A&A

One of the fundamental aspects of statistical behaviour in many-body systems is exponential divergence of neighbouring orbits, which is often discussed in terms of Liapounov exponents. Here we study this topic for the classical gravitational N-body problem. The application we have in mind is to old stellar systems such as globular star clusters, where N~10^6, and so we concentrate on spherical, centrally concentrated systems with total energy E<0. Hitherto no connection has been made between the time scale for divergence (denoted here by t_e) and the two-body relaxation time scale (t_r), even though both may be calculated by consideration of two-body encounters. In this paper we give a simplified model showing that divergence in phase space is initially roughly exponential, on a timescale proportional to the crossing time (defined as a mean time for a star to cross from one side of the system to another). In this phase t_e is much less than t_r, if N is not too small (i.e. N is much more than about 30). After several e-folding times, the model shows that the divergence slows down. Thereafter the divergence (measured by the energies of the bodies) varies with time as t^{1/2}, on a timescale nearly proportional to the familiar two-body relaxation timescale, i.e. t_e ~ t_r in this phase. These conclusions are illustrated by numerical results., Comment: 19 pages, 3 figures, accepted by J. Stat. Phys; new figures, numerical results and references; improved text; fig.1 at low resolution in this version

In this paper a cluster is modelled as a smooth potential (due to the cluster stars) plus the steady tidal field of the Galaxy. In this model there is a minimum energy below which stars cannot escape. Above this energy, however, the time scale on which stars escape varies with the orbital parameters of the star (mainly its energy) in a way which we attempt to quantify, with both theoretical arguments and computer simulations. Within the limitations of the model we show that the time scale is long enough to complicate the interpretation of full N-body simulations of clusters, and that stars above the escape energy may remain bound to the cluster for about a Hubble time., Comment: 15 pages, 11 figures, plain TeX, submitted to MNRAS

We consider the use of N-body simulations for studying the evolution of rich star clusters (i.e. globular clusters). The dynamical processes included in this study are restricted to gravitational (point-mass) interactions, the steady tidal field of a galaxy, and instantaneous mass loss resulting from stellar evolution. With evolution driven by these mechanisms, it is known that clusters fall roughly into two broad classes: those that dissipate promptly in the tidal field, as a result of mass loss, and those that survive long enough for their evolution to become dominated by two-body relaxation. The time scales of the processes we consider scale in different ways with the number of stars in the simulation, and the main aim of the paper is to suggest how the scaling of a simulation should be done so that the results are representative of the evolution of a `real' cluster. We investigate three different ways of scaling time. One of these is appropriate to the first type of cluster, i.e. those that dissipate rapidly, and similarly a second scaling is appropriate only to the second (relaxation-dominated) type. We also develop a hybrid scaling which is a satisfactory compromise for both types of cluster. Finally we present evidence that the widely used Fokker-Planck method produces models which are in good agreement with N-body models of those clusters which are relaxation dominated, at least for N-body models with several thousand particles, but that the Fokker-Planck models evolve too fast for clusters which dissipate promptly., Comment: 24 pages, 3 figures, MNRAS, in press

We review recent progress in computational methods for studying the dynamical evolution of large star clusters. After describing some highlights of a new comparison of different models, including N-body models, we mention in detail recent developments in three codes which allow models with anisotropic distributions of velocity to be studied: (i) a Fokker-Planck code using finite differences, (ii) a Monte Carlo code and (iii) a gaseous model code., Comment: 6 pages, postscript, etc., also available at http://www.maths.ed.ac.uk/people/douglas/preprints.html To appear in Andersen J., ed, Highlights of Astronomy Vol. 11. Kluwer, Dordrecht. Pagination differs from that in published version

The simulation of rich star clusters presents challenging problems of several kinds, including the design of suitable hardware and software, and numerous theoretical problems in stellar dynamics and stellar physics. Great progress has been made possible in recent years through the widespread use of GRAPE hardware. Simulations are, however, still too small to be applied to real star clusters without scaling. How this is done is partly an issue of stellar dynamics, and it has thrown into focus a number of fundamental theoretical problems in this field., Comment: 10 pages, postscript, also available at http://www.maths.ed.ac.uk/people/douglas/preprints.html To appear in T. Ebisuzaki, ed, Proc. Int. Symp. on Supercomputing - New Horizon of Computational Science, Tokyo, Japan, 1-3 September, 1997. Kluwer, Dordrecht

In this paper we show the results of a large set of N-body simulations modelling the evolution of globular clusters driven by relaxation,stellar evolution,disk shocking and including the effects of the tidal field of the Galaxy. We investigate the evolution of multi-mass models with a power-law initial mass function (IMF) starting with different initial masses, concentrations, slopes of the IMF and located at different galactocentric distances. We show to what extent the effects of the various evolutionary processes alter the shape of the IMF and to what extent these changes depend on the position of the cluster in the Galaxy. Both the changes in the global mass function and in the local one (measured at different distances from the cluster center) are investigated showing whether and where the local mass function keeps memory of the IMF and where it provides a good indication of the current global mass function. The evolution of the population of white dwarfs is also followed in detail and we supply an estimate of the fraction of the current value of the total mass expected to be in white dwarfs depending on the main initial conditions for the cluster (mass and position in the Galaxy).Simple analytical expression by which it is possible to calculate the main quantities of interest (total mass, fraction of white dwarfs, slope of the mass function) at any time t for a larger number of different initial conditions than those investigated numerically have been derived., Comment: 20 pages LaTeX (MNRAS style), 23 figures, MNRAS (in press)

Galactic globular clusters, which are ancient building blocks of our Galaxy, represent a very interesting family of stellar systems in which some fundamental dynamical processes have taken place on time scales shorter than the age of the universe. In contrast with galaxies, these clusters represent unique laboratories for learning about two-body relaxation, mass segregation from equipartition of energy, stellar collisions, stellar mergers, and core collapse. In the present review, we summarize the tremendous developments, as much theoretical as observational, that have taken place during the last two decades, and which have led to a quantum jump in our understanding of these beautiful dynamical systems., Comment: A review of 171 pages (TeX, no special macros) in press in The Astronomy and Astrophysics Review. Only 7 figures available in postscript form

We present the first comprehensive fitting formula for exchange reactions of arbitrary mass ratios. In a comparison with numerical results, this expression is shown to be accurate in the hard binary limit to within 25\% for most mass ratios. The result will be useful in forming quantitative estimates for the branching ratios of various exchange reactions in astrophysical applications. For example, it can be used to construct quantitative formation scenarios for unusual objects in globular clusters, such as binaries containing a pulsar., Comment: 25 pages, TeX + 2 ps-figures, to appear in ApJ

This paper generalises King's models for tidally truncated star clusters by including approximately the non-spherical symmetry of the tidal field and the resulting non-spherical distortion of the cluster., Comment: 22 pages