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Douglas C. Heggie

Publications and source records attributed to Douglas C. Heggie.

At least 37 records · Page 2Linked to original sources
arXiv2013

On black hole sub-systems in idealized nuclear star clusters

Recent observational evidence, numerical simulations and theoretical arguments seem to indicate that stellar mass black holes (BH) persist in stellar systems such as globular star clusters for much longer than previously thought. Previously, theory implied that the BH would segregate into a compact system with short dynamical time scales, and that the BH would escape long before the present. But stellar systems can exist in a state of balanced evolution, where the energy generated in the core is regulated by the process of two-body relaxation in the bulk of the system. If the system has a centrally concentrated BH subsystem and there is no massive central BH, then the energy is generated by three-body encounters in the core of this subsystem. Therefore, in balanced evolution, the evolution of the BH subsystem is regulated by the much longer time scales of the host system. In the present paper the implications of these results for idealized nuclear star clusters (NSC) are discussed. Though previous theory implied that BH would be almost absent from many NSC -- those with relatively short dynamical time scales -- it is argued here that, based on the results of idealized models, many such NSC could still be host to substantial BH subsystems.

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arXiv2013

Dynamical evolution of black hole sub-systems in idealised star clusters

In this paper, globular star clusters which contain a sub-system of stellar-mass black holes (BH) are investigated. This is done by considering two-component models, as these are the simplest approximation of more realistic multi-mass systems, where one component represents the BH population and the other represents all the other stars. These systems are found to undergo a long phase of evolution where the centre of the system is dominated by a dense BH sub-system. After mass segregation has driven most of the BH into a compact sub-system, the evolution of the BH sub-system is found to be influenced by the cluster in which it is contained. The BH sub-system evolves in such a way as to satisfy the energy demands of the whole cluster, just as the core of a one component system must satisfy the energy demands of the whole cluster. The BH sub-system is found to exist for a significant amount of time. It takes approximately 10t_{rh,i}, where t_{rh,i} is the initial half-mass relaxation time, from the formation of the compact BH sub-system up until the time when 90% of the sub-system total mass is lost (which is of order 10^{3} times the half-mass relaxation time of the BH sub-system at its time of formation). Based on theoretical arguments the rate of mass loss from the BH sub-system (\dot{M}_2) is predicted to be -(beta*zeta*M)/(alpha*t_{rh}), where M is the total mass, t_{rh} is the half-mass relaxation time, and alpha, beta, zeta are three dimensionless parameters (see Section 2 for details). An interesting consequence of this is that the rate of mass loss from the BH sub-system is approximately independent of the stellar mass ratio (m_2/m_1) and the total mass ratio (M_2/M_1) (in the range m_2/m_1 >~ 10 and M_2/M_1 ~ 10^{-2}, where m_1, m_2 are the masses of individual low-mass and high-mass particles respectively, and M_1, M_2 are the corresponding total masses).

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arXiv2012

Gravothermal oscillations in multi-component models of star clusters

In this paper, gravothermal oscillations are investigated in multi-component star clusters which have power law initial mass functions (IMF). For the power law IMFs, the minimum masses ($m_{min}$) were fixed and three different maximum stellar masses ($m_{max}$) were used along with different power-law exponents ($α$) ranging from 0 to -2.35 (Salpeter). The critical number of stars at which gravothermal oscillations first appear with increasing $N$ was found using the multi-component gas code SPEDI. The total mass ($M_{tot}$) is seen to give an approximate stability condition for power law IMFs with fixed values of $m_{max}$ and $m_{min}$ independent of $α$. The value $M_{tot}/m_{max} \simeq 12000$ is shown to give an approximate stability condition which is also independent of $m_{max}$, though the critical value is somewhat higher for the steepest IMF that was studied. For appropriately chosen cases, direct N-body runs were carried out in order to check the results obtained from SPEDI. Finally, evidence of the gravothermal nature of the oscillations found in the N-body runs is presented.

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arXiv2012

The Tidal Tails of 47 Tucanae

The Galactic globular cluster 47 Tucanae (47 Tuc) shows a rare increase in its velocity dispersion profile at large radii, indicative of energetic, yet bound, stars at large radii dominating the velocity dispersion and, potentially, of ongoing evaporation. Escaping stars will form tidal tails, as seen with several Galactic globular clusters, however, the tidal tails of 47 Tuc are yet to be uncovered. We model these tails of 47 Tuc using the most accurate input data available, with the specific aim of determining their locations, as well as the densities of the epicyclic overdensities within the tails. The overdensities from our models show an increase of 3-4% above the Galactic background and, therefore, should be easily detectable using matched filtering techniques. We find that the most influential parameter with regard to both the locations and densities of the epicyclic overdensities is the Heliocentric distance to the cluster. Hence, uncovering these tidal features observationally will contribute greatly to the ongoing problem of determining the distance to 47 Tuc, tightly constraining the distance of the cluster independent of other methods. Using our streakline method for determining the locations of the tidal tails and their overdensities, we show how, in principle, the shape and extent of the tidal tails of any Galactic globular cluster can be determined without resorting to computationally expensive N-body simulations.

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arXiv2011

More on the structure of tidal tails

We investigate the epicyclic motion of stars escaping from star clusters. Using streaklines, we visualise the path of escaping stars and show how epicyclic motion leads to over- and underdensities in tidal tails of star clusters moving on circular and eccentric orbits about a galaxy. Additionally, we investigate the effect of the cluster mass on the tidal tails, by showing that their structure is better matched when the perturbing effect of the cluster mass is included. By adjusting streaklines to results of N-body computations we can accurately and quickly reproduce all observed substructure, especially the streaky features often found in simulations which may be interpreted in observations as multiple tidal tails. Hence, we can rule out tidal shocks as the origin of such substructures. Finally, from the adjusted streakline parameters we can verify that for the star clusters we studied escape mainly happens from the tidal radius of the cluster, given by x_L = (GM/(Ω^2-\partial^2Φ/\partial R^2))^{1/3}. We find, however, that there is another limiting radius, the "edge" radius, which gives the smallest radius from which a star can escape during one cluster orbit about the galaxy. For eccentric cluster orbits the edge radius shrinks with increasing orbital eccentricity (for fixed apocentric distance) but is always significantly larger than the respective perigalactic tidal radius. In fact, the edge radii of the clusters we investigated, which are extended and tidally filling, agree well with their (fitted) King radii, which may indicate a fundamental connection between these two quantities.

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arXiv2011

Gravothermal oscillations in two-component models of star clusters

In this paper, gravothermal oscillations are investigated in two-component clusters with a range of different stellar mass ratios and total component mass ratios. The critical number of stars at which gravothermal oscillations first appeared is found using a gas code. The nature of the oscillations is investigated and it is shown that the oscillations can be understood by focusing on the behaviour of the heavier component, because of mass segregation. It is argued that, during each oscillation, the re-collapse of the cluster begins at larger radii while the core is still expanding. This re-collapse can halt and reverse a gravothermally driven expansion. This material outside the core contracts because it is losing energy both to the cool expanding core and to the material at larger radii. The core collapse times for each model are also found and discussed. For an appropriately chosen case, direct N -body runs were carried out, in order to check the results obtained from the gas model, including evidence of the gravothermal nature of the oscillations and the temperature inversion that drives the expansion.

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arXiv2011

The life cycle of star cluster in a tidal field

The evolution of globular clusters due to 2-body relaxation results in an outward flow of energy and at some stage all clusters need a central energy source to sustain their evolution. Henon provided the insight that we do not need to know the details of the energy production in order to understand the relaxation-driven evolution of the cluster, at least outside the core. He provided two self-similar solutions for the evolution of clusters based on the view that the cluster as a whole determines the amount of energy that is produced in the core: steady expansion for isolated clusters, and homologous contraction for clusters evaporating in a tidal field. We combine these models: the half-mass radius increases during the first half of the evolution, and decreases in the second half; while the escape rate approaches a constant value set by the tidal field. We refer to these phases as `expansion dominated' and `evaporation dominated'. These simple analytical solutions immediately allow us to construct evolutionary tracks and isochrones in terms of cluster half-mass density, cluster mass and galacto-centric radius. From a comparison to the Milky Way globular clusters we find that roughly 1/3 of them are in the second, evaporation-dominated phase and for these clusters the density inside the half-mass radius varies with the galactocentric distance R as rho_h ~ 1/R^2. The remaining 2/3 are still in the first, expansion-dominated phase and their isochrones follow the environment-independent scaling rho_h ~ M^2; that is, a constant relaxation time-scale. We find substantial agreement between Milky Way globular cluster parameters and the isochrones, which suggests that there is, as Henon suggested, a balance between the flow of energy and the central energy production for almost all globular clusters.

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arXiv2011

Problems of Collisional Stellar Dynamics

The discovery of dynamical friction was Chandrasekhar's best known contribution to the theory of stellar dynamics, but his work ranged from the few-body problem to the limit of large N (in effect, galaxies). Much of this work was summarised in the text "Principles of Stellar Dynamics" (Chandrasekhar 1942, 1960), which ranges from a precise calculation of the time of relaxation, through a long analysis of galaxy models, to the behaviour of star clusters in tidal fields. The later edition also includes the work on dynamical friction and related issues. In this review we focus on progress in the collisional aspects of these problems, i.e. those where few-body interactions play a dominant role, and so we omit further discussion of galaxy dynamics. But we try to link Chandrasekhar's fundamental discoveries in collisional problems with the progress that has been made in the 50 years since the publication of the enlarged edition.

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arXiv2010

The effects of fly-bys on planetary systems

Most of the observed extrasolar planets are found on tight and often eccentric orbits. The high eccentricities are not easily explained by planet-formation models, which predict that planets should be on rather circular orbits. Here we explore whether fly-bys involving planetary systems with properties similar to those of the gas giants in the solar system, can produce planets with properties similar to the observed planets. Using numerical simulations, we show that fly-bys can cause the immediate ejection of planets, and sometimes also lead to the capture of one or more planets by the intruder. More common, however, is that fly-bys only perturb the orbits of planets, sometimes leaving the system in an unstable state. Over time-scales of a few million to several hundred million years after the fly-by, this perturbation can trigger planet-planet scatterings, leading to the ejection of one or more planets. For example, in the case of the four gas giants of the solar system, the fraction of systems from which at least one planet is ejected more than doubles in 10^8 years after the fly-by. The remaining planets are often left on more eccentric orbits, similar to the eccentricities of the observed extrasolar planets. We combine our results of how fly-bys effect solar-system-like planetary systems, with the rate at which encounters in young stellar clusters occur. For example, we measure the effects of fly-bys on the four gas giants in the solar system. We find, that for such systems, between 5 and 15 per cent suffer ejections of planets in 10^8 years after fly-bys in typical open clusters. Thus, encounters in young stellar clusters can significantly alter the properties of any planets orbiting stars in clusters. As a large fraction of stars which populate the solar neighbourhood form in stellar clusters, encounters can significantly affect the properties of the observed extrasolar planets.

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arXiv2010

Monte Carlo Simulations of Star Clusters - VII. The globular cluster 47 Tuc

We describe Monte Carlo models for the dynamical evolution of the massive globular cluster 47 Tuc (NGC 104). 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 surface brightness and density profiles, the velocity dispersion profile, the luminosity function in two fields, and the acceleration of pulsars. Our models appear to require a relatively steep initial mass function for stars above about turnoff, with an index of about 2.8 (where the Salpeter mass function has an index of 2.35), and a relatively flat initial mass function (index about 0.4) for the lower main sequence. According to the model, the current mass is estimated at 0.9 million solar masses, of which about 34% consists of remnants. We find that primordial binaries are gradually taking over from mass loss by stellar evolution as the main dynamical driver of the core. Despite the high concentration of the cluster, core collapse will take at least another 20Gyr.

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arXiv2009

1Gyr in the Life of the Globular Cluster NGC 6397

M4 and NGC 6397 are two very similar galactic globular clusters, which differ mainly in their surface brightness profile. M4 has a classic King-like profile, whereas NGC 6397 has a more concentrated profile, which is often interpreted as that of a post-core collapse cluster. Heggie & Giersz (2008), however, found that M4 is also a post-core collapse cluster, and Giersz & Heggie (2009) concluded that the main reason for the difference between the two surface brightness profiles is fluctuations. This conclusion was reached on the basis of Monte Carlo models, however, and in the present Letter we verify that similar fluctuations occur in N-body models. The models were initialised by generating initial conditions from the Monte Carlo model of NGC 6397 at the simulated age of 12Gyr, and one was followed for 1Gyr. The new models help to clarify the nature of the fluctuations, which have the nature of semi-regular oscillations with a time scale of order 10^8yr. They are influenced by the dynamical role which is played by primordial binaries in the evolution of the core.

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arXiv2008

On the structure of tidal tails

We examine the longitudinal distribution of the stars escaping from a cluster along tidal tails. Using both theory and simulations, we show that, even in the case of a star cluster in a circular galactic orbit, when the tide is steady, the distribution exhibits maxima at a distance of many tidal radii from the cluster.

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arXiv2008

Monte Carlo Simulations of Star Clusters - IV. Calibration of the Monte Carlo Code and Comparison with Observations for the Open Cluster M67

We outline the steps needed in order to incorporate the evolution of single and binary stars into a particular Monte Carlo code for the dynamical evolution of a star cluster. We calibrate the results against N-body simulations, and present models for the evolution of the old open cluster M67 (which has been studied thoroughly in the literature with N-body techniques). The calibration is done by choosing appropriate free code parameters. We describe in particular the evolution of the binary, white dwarf and blue straggler populations, though not all channels for blue straggler formation are represented yet in our simulations. Calibrated Monte Carlo runs show good agreement with results of N-body simulations not only for global cluster parameters, but also for e.g. binary fraction, luminosity function and surface brightness. Comparison of Monte Carlo simulations with observational data for M67 shows that is possible to get reasonably good agreement between them. Unfortunately, because of the large statistical fluctuations of the numerical data and uncertainties in the observational data the inferred conclusions about the cluster initial conditions are not firm.

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arXiv2007

Modeling Dense Stellar Systems

Black holes and neutron stars present extreme forms of matter that cannot be created as such in a laboratory on Earth. Instead, we have to observe and analyze the experiments that are ongoing in the Universe. The most telling observations of black holes and neutron stars come from dense stellar systems, where stars are crowded close enough to each other to undergo frequent interactions. It is the interplay between black holes, neutron stars and other objects in a dense environment that allows us to use observations to draw firm conclusions about the properties of these extreme forms of matter, through comparisons with simulations. The art of modeling dense stellar systems through computer simulations forms the main topic of this review.

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arXiv2007

The core radius of a star cluster containing a massive black hole

(Simplified) We present a theoretical framework which establishes how the core radius of a star cluster varies with the mass of an assumed central black hole. Our result is that the ratio of core to half-mass radius varies as the 3/4 power of the ratio of the black hole to cluster mass, when the system is well relaxed. The theory compares favourably with a number of simulations of this problem, which extend to black hole masses of order 10% of the cluster mass. Though strictly limited as yet to clusters with stars of equal mass, our conclusion strengthens the view that clusters with large core radii are the most promising candidates in which to find a massive black hole.

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arXiv2007

Gravitational Scattering

We review some modern applications of the theory of few-body encounters between binaries and single stars. Then we focus on a new treatment of adiabatic encounters, in a regime which is of importance in encounters between a star and a planetary system in a star cluster.

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arXiv2007

Exact equilibria of a stellar system in a linearised tidal field

We study the motion of stars in a star cluster which revolves in a circular orbit about its parent galaxy. The star cluster is modelled as an ellipsoid of uniform spatial density. We exhibit two 2-parameter families of self-consistent equilibrium models in which the velocity at each point is confined to a line in velocity space. We exhibit the link between this problem and that of a uniform rotating ellipsoidal galaxy. With minimal adaptation, Freeman's bar models yield a third family.

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