arXiv subjects

Douglas C. Heggie

Publications and source records attributed to Douglas C. Heggie.

At least 19 recordsLinked to original sources

arXiv2026

Discrepancies between Chandrasekhar's theory of relaxation and $N$-body simulations

Globular clusters are systems which are known to be particularly well described by two-body relaxation. In recent decades many studies have shown that Chandrasekhar's orbit-averaged theory is able to reproduce many features of numerical simulations. However, it has been claimed that differences between the theory and simulation remain, such as an amplitude mismatch of the rate of change of the distribution function. In this paper we compare the theoretical predictions of Chandrasekhar's theory for anisotropic clusters to precise $N$-body data. We show that more careful $N$-body measurements are able to reduce the claimed mismatch. Nevertheless, we observe a dependency of the remaining mismatch on both position and anisotropy. While the dependence on anisotropy may be understood qualitatively on theoretical grounds, the radial dependence implies that spatial inhomogeneities, and therefore collective effects, may become unavoidable to resolve the residual mismatch between theory and simulations.

astro-ph.GA↗

arXiv2024

Dynamics of star clusters with tangentially anisotropic velocity distribution

Recent high-precision observations with HST and Gaia enabled new investigations of the internal kinematics of star clusters (SCs) and the dependence of kinematic properties on the stellar mass. These studies raised new questions about the dynamical evolution of self-gravitating stellar systems. We aim to develop a more complete theoretical understanding of how various kinematical properties of stars affect the global dynamical development of their host SCs. We perform N-body simulations of SCs with isotropic, radially anisotropic and tangentially anisotropic initial velocity distributions. We also study the effect of an external Galactic tidal field. We find three main results. First, compared to the conventional, isotropic case, the relaxation processes are accelerated in the tangentially anisotropic models and, in agreement with our previous investigations, slower in the radially anisotropic ones. This leads to, e.g., more rapid mass segregation in the central regions of the tangential models or their earlier core collapse. Second, although all SCs become isotropic in the inner regions after several relaxation times, we observe differences in the anisotropy profile evolution in the outer cluster regions - all tidally filling models gain tangential anisotropy there while the underfilling models become radially anisotropic. Third, we observe different rates of evolution towards energy equipartition (EEP). While all SCs evolve towards EEP in their inner regions (regardless of the filling factor), the outer regions of the tangentially anisotropic and isotropic models are evolving to an "inverted" EEP (i.e., the high-mass stars having higher velocity dispersion than the low-mass ones). The extent (both spatial and temporal) of this inversion can be attributed to the initial velocity anisotropy - it grows with increasing tangential anisotropy and decreases as the radial anisotropy rises.

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arXiv2023

Punctuated Chaos and Indeterminism in Self-gravitating Many-body Systems

Dynamical chaos is a fundamental manifestation of gravity in astrophysical, many-body systems. The spectrum of Lyapunov exponents quantifies the associated exponential response to small perturbations. Analytical derivations of these exponents are critical for understanding the stability and predictability of observed systems. This essay presents a new model for chaos in systems with eccentric and crossing orbits. Here, exponential divergence is not a continuous process but rather the cumulative effect of an ever-increasing linear response driven by discrete events at regular intervals, i.e., punctuated chaos. We show that long-lived systems with punctuated chaos can magnify Planck length perturbations to astronomical scales within their lifetime, rendering them fundamentally indeterministic.

astro-ph.IM↗

arXiv2022

The Effect of Stars on the Dark Matter Spike Around a Black Hole: A Tale of Two Treatments

We revisit the role that gravitational scattering off stars plays in establishing the steady-state distribution of collisionless dark matter (DM) around a massive black hole (BH). This is a physically interesting problem that has potentially observable signatures, such as $γ-$rays from DM annihilation in a density spike. The system serves as a laboratory for comparing two different dynamical approaches, both of which have been widely used: a Fokker-Planck treatment and a two-component conduction fluid treatment. In our Fokker-Planck analysis we extend a previous analytic model to account for a nonzero flux of DM particles into the BH, as well as a cut-off in the distribution function near the BH due to relativistic effects or, further out, possible DM annihilation. In our two-fluid analysis, following an approximate analytic treatment, we recast the equations as a "heated Bondi accretion" problem and solve the equations numerically without approximation. While both the Fokker-Planck and two-fluid methods yield basically the same DM density and velocity dispersion profiles away from the boundaries in the spike interior, there are other differences, especially the determination of the DM accretion rate. We discuss limitations of the two treatments, including the assumption of an isotropic velocity dispersion.

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arXiv2021

$l=1$: Weinberg's weakly damped mode in an $N$-body model of a spherical stellar system

Spherical stellar systems such as King models, in which the distribution function is a decreasing function of energy and depends on no other invariant, are stable in the sense of collisionless dynamics. But Weinberg showed, by a clever application of the matrix method of linear stability, that they may be nearly unstable, in the sense of possessing {\sl weakly} damped modes of oscillation. He also demonstrated the presence of such a mode in an $N$-body model by endowing it with initial conditions generated from his perturbative solution. In the present paper we provide evidence for the presence of this same mode in $N$-body simulations of the King $W_0 = 5$ model, in which the initial conditions are generated by the usual Monte Carlo sampling of the King distribution function. It is shown that the oscillation of the density centre correlates with variations in the structure of the system out to a radius of about 1 virial radius, but anticorrelates with variations beyond that radius. Though the oscillations appear to be continually reexcited (presumably by the motions of the particles) we show by calculation of power spectra that Weinberg's estimate of the period (strictly, $2π$ divided by the real part of the eigenfrequency) lies within the range where the power is largest. In addition, however, the power spectrum displays another very prominent feature at shorter periods, around 5 crossing times.

astro-ph.GA↗

arXiv2021

The Kinematic Richness of Star Clusters - II. Stability of Spherical Anisotropic Models with Rotation

We study the bar instability in collisionless, rotating, anisotropic, stellar systems, using N-body simulations and also the matrix technique for calculation of modes with the perturbed collisionless Boltzmann equation. These methods are applied to spherical systems with an initial Plummer density distribution, but modified kinematically in two ways: the velocity distribution is tangentially anisotropic, using results of Dejonghe, and the system is set in rotation by reversing the velocities of a fraction of stars in various regions of phase space, a la Lynden-Bell. The aim of the N-body simulations is first to survey the parameter space, and, using those results, to identify regions of phase space (by radius and orbital inclination) which have the most important influence on the bar instability. The matrix method is then used to identify the resonant interactions in the system which have the greatest effect on the growth rate of a bar. Complementary series of N-body simulations examine these processes in relation to the evolving frequency distribution and the pattern speed. Finally, the results are synthesised with an existing theoretical framework, and used to consider the old question of constructing a stability criterion.

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arXiv2019

Spherical models of star clusters with potential escapers

An increasing number of observations of the outer regions of globular clusters (GCs) have shown a flattening of the velocity dispersion profile and an extended surface density profile. Formation scenarios of GCs can lead to different explanations of these peculiarities, therefore the dynamics of stars in the outskirts of GCs are an important tool in tracing back the evolutionary history and formation of star clusters. One possible explanation for these features is that GCs are embedded in dark matter halos. Alternatively, these features are the result of a population of energetically unbound stars that can be spatially trapped within the cluster, known as potential escapers (PEs). We present a prescription for the contribution of these energetically unbound members to a family of self-consistent, distribution function-based models, which, for brevity, we call the Spherical Potential Escapers Stitched (SPES) models. We show that, when fitting to mock data of bound and unbound stars from an N-body model of a tidally-limited star cluster, the SPES models correctly reproduce the density and velocity dispersion profiles up to the Jacobi radius, and they are able to recover the value of the Jacobi radius itself to within 20%. We also provide a comparison to the number density and velocity dispersion profiles of the Galactic cluster 47 Tucanae. Such a case offers a proof of concept that an appropriate modelling of PEs is essential to accurately interpret Gaia data in the outskirts of GCs, and, in turn, to formulate meaningful present-day constraints for GC formation scenarios in the early universe.

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arXiv2019

Mapping the stability of stellar rotating spheres via linear response theory

Rotation is ubiquitous in the Universe, and recent kinematic surveys have shown that early type galaxies and globular clusters are no exception. Yet the linear response of spheroidal rotating stellar systems has seldom been studied. This paper takes a step in this direction by considering the behaviour of spherically symmetric systems with differential rotation. Specifically, the stability of several sequences of Plummer spheres is investigated, in which the total angular momentum, as well as the degree and flavour of anisotropy in the velocity space are varied. To that end, the response matrix method is customised to spherical rotating equilibria. The shapes, pattern speeds and growth rates of the systems' unstable modes are computed. Detailed comparisons to appropriate N-body measurements are also presented. The marginal stability boundary is charted in the parameter space of velocity anisotropy and rotation rate. When rotation is introduced, two sequences of growing modes are identified corresponding to radially and tangentially-biased anisotropic spheres respectively. For radially anisotropic spheres, growing modes occur on two intersecting surfaces (in the parameter space of anisotropy and rotation), which correspond to fast and slow modes, depending on the net rotation rate. Generalised, approximate stability criteria are finally presented.

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arXiv2017

The Kinematic Richness of Star Clusters I. Isolated Spherical Models with Primordial Anisotropy

We investigate the dynamical evolution of isolated equal-mass star cluster models by means of direct N-body simulations, primarily focusing on the effects of the presence of primordial anisotropy in the velocity space. We found evidence of the existence of a monotonic relationship between the moment of core collapse and the amount and flavour of anisotropy in the stellar system. Specifically, equilibria characterised by the same initial structural properties (Plummer density profile) and with different degrees of tangentially-biased (radially-biased) anisotropy, reach core collapse earlier (later) than isotropic models. We interpret this result in light of an accelerated (delayed) phase of the early evolution of collisional stellar systems "anisotropic-response"), which we have characterised both in terms of the evolution of the velocity moments and of a fluid model of two-body relaxation. For the case of the most tangentially anisotropic model the initial phase of evolution involves a catastrophic collapse of the inner part of the system which continues until an isotropic velocity distribution is reached. This study represents a first step towards a comprehensive investigation of the role played by kinematic richness in the long-term dynamical evolution of collisional systems.

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arXiv2017

An Approximate Analytic Model of a Star Cluster with Potential Escapers

In the context of a star cluster moving on a circular galactic orbit, a "potential escaper" is a cluster star that has orbital energy greater than the escape energy, and yet is confined within the Jacobi radius of the stellar system. On the other hand analytic models of stellar clusters typically have a truncation energy equal to the cluster escape energy, and therefore explicitly exclude these energetically unbound stars. Starting from the landmark analysis performed by Henon of periodic orbits of the circular Hill equations, we present a numerical exploration of the population of "non-escapers", defined here as those stars which remain within two Jacobi radii for several galactic periods, with energy above the escape energy. We show that they can be characterised by the Jacobi integral and two further approximate integrals, which are based on perturbation theory and ideas drawn from Lidov-Kozai theory. Finally we use these results to construct an approximate analytic model that includes a phase space description of a population resembling that of potential escapers, in addition to the usual bound population.

astro-ph.GA↗

arXiv2016

The Origin of Chaos in the Orbit of Comet 1P/Halley

According to Munoz-Gutierrez et al. (2015) the orbit of comet 1P/Halley is chaotic with a surprisingly small Lyapunov time scale of order its orbital period. In this work we analyse the origin of chaos in Halley's orbit and the growth of perturbations, in order to get a better understanding of this unusually short time scale. We perform N-body simulations to model Halley's orbit in the Solar System and measure the separation between neighbouring trajectories. To be able to interpret the numerical results, we use a semi-analytical map to demonstrate different growth modes, i.e. linear, oscillatory or exponential, and transitions between these modes. We find the Lyapunov time scale of Halley's orbit to be of order 300 years, which is significantly longer than previous estimates in the literature. This discrepancy could be due to the different methods used to measure the Lyapunov time scale. A surprising result is that next to Jupiter, also encounters with Venus contribute to the exponential growth in the next 3000 years. Finally, we note an interesting application of the sub-linear, oscillatory growth mode to an ensemble of bodies moving through the Solar System. Whereas in the absence of encounters with a third body the ensemble spreads out linearly in time, the accumulation of weak encounters can increase the lifetime of such systems due to the oscillatory behaviour.

astro-ph.EP↗

arXiv2015

Evolution of star clusters on eccentric orbits

We study the evolution of star clusters on circular and eccentric orbits using direct $N$-body simulations. We model clusters with initially $N=8{\rm k}$ and $N=16{\rm k}$ single stars of the same mass, orbiting around a point-mass galaxy. For each orbital eccentricity that we consider, we find the apogalactic radius at which the cluster has the same lifetime as the cluster with the same $N$ on a circular orbit. We show that then, the evolution of bound particle number and half-mass radius is approximately independent of eccentricity. Secondly, when we scale our results to orbits with the same semi-major axis, we find that the lifetimes are, to first order, independent of eccentricity. When the results of Baumgardt and Makino for a singular isothermal halo are scaled in the same way, the lifetime is again independent of eccentricity to first order, suggesting that this result is independent of the Galactic mass profile. From both sets of simulations we empirically derive the higher order dependence of the lifetime on eccentricity. Our results serve as benchmark for theoretical studies of the escape rate from clusters on eccentric orbits. Finally, our results can be useful for generative models for cold streams and cluster evolution models that are confined to spherical symmetry and/or time-independent tides, such as Fokker-Planck models, Monte Carlo models, and (fast) semi-analytic models.

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arXiv2015

Neutron star natal kicks and the long-term survival of star clusters

We investigate the dynamical evolution of a star cluster in an external tidal field by using N-body simulations, with focus on the effects of the presence or absence of neutron star natal velocity kicks.We show that, even if neutron stars typically represent less than 2% of the total bound mass of a star cluster, their primordial kinematic properties may affect the lifetime of the system by up to almost a factor of four. We interpret this result in the light of two known modes of star cluster dissolution, dominated by either early stellar evolution mass loss or two-body relaxation. The competition between these effects shapes the mass loss profile of star clusters, which may either dissolve abruptly ("jumping"), in the pre-core-collapse phase, or gradually ("skiing"), after having reached core collapse.

astro-ph.GA↗

arXiv2014

Michel Henon's contributions to collisional stellar systems

The theory of star cluster dynamics was a major topic in Hénon's early research career. Here we summarise his contributions under three headings: (i) the Monte Carlo method, (ii) homological evolution of star clusters, and (iii) escape from star clusters. In each case we also trace some aspects of how Hénon's contributions have been developed or applied in subsequent decades up to the present. We also propose that Hénon's work be commemorated by adopting the names "Hénon units" and "Hénon's Principle".

astro-ph.GA↗

arXiv2014

Towards an N-body Model for the Globular Cluster M4

This paper describes an N-body model for the dynamical evolution of the nearby globular cluster M4. The initial conditions, with N = 484710 particles, were generated from a published study of this cluster with a Monte Carlo code. With the Monte Carlo code, these initial conditions led, after 12 Gyr of dynamical and stellar evolution, to a model which resembles M4 in terms of its surface brightness and velocity dispersion profiles, and its local luminosity function. Though the N-body model reported here is marred by some errors, its evolution can be compared with that of the published Monte Carlo model, with a result from the synthetic evolution code EMACSS, and with M4 itself.

astro-ph.GA↗

arXiv2014

MOCCA code for star cluster simulations - III. Stellar-mass black holes in the globular cluster M22

Using a Monte Carlo code, we construct a dynamic evolutionary model of the Galactic globular cluster M22 (NGC6656). The initial conditions are chosen so that, after about 12Gyr of stellar and dynamical evolution, the model is an approximate fit to the surface brightness and velocity dispersion profiles of the cluster, to its mass function, and to the current binary fraction. Depending on the distribution of black hole natal kicks, we predict that the present-day population of stellar-mass black holes ranges from about 40 (no kicks) down to essentially zero (kicks distributed like those of neutron stars). Provided that natal kicks do not eject all new black holes, it is suggested that clusters with a present-day half-mass relaxation time above about 1Gyr are the ones that may still retain an appreciable population of black holes.

astro-ph.GA↗

arXiv2013

MOCCA Code for Star Cluster Simulations - II. Comparison with N-body Simulations

We describe a major upgrade of a Monte Carlo code which has previously been used for many studies of dense star clusters. We outline the steps needed in order to calibrate the results of the new Monte Carlo code against $N$-body simulations for large $N$ systems, up to $N=200000$. The new version of the Monte Carlo code (called MOCCA), in addition to the features of the old version, incorporates the direct Fewbody integrator (Fregeau et al. 2004) for three- and four-body interactions, and a new treatment of the escape process based on Fukushige & Heggie (2000). Now stars which fulfil the escape criterion are not removed immediately, but can stay in the system for a certain time which depends on the excess of the energy of a star above the escape energy. They are called potential escapers. With the addition of the Fewbody integrator the code can follow all interaction channels which are important for the rate of creation of various types of objects observed in star clusters, and ensures that the energy generation by binaries is treated in a manner similar to the $N$-body model. There are at most three new parameters which have to be adjusted against $N$-body simulations for large $N$: two (or one, depending on the chosen approach) connected with the escape process, and one responsible for the determination of the interaction probabilities. The values adopted for the free parameters have at most a weak dependence on $N$. They allow MOCCA to reproduce $N$-body results with reasonable precision, not only for the rate of cluster evolution and the cluster mass distribution, but also for the detailed distributions of mass and binding energy of binaries. Additionally, the code can follow the rate of formation of blue stragglers and black hole - black hole binaries.

astro-ph.GA↗

arXiv2013

Few-Body Modes of Binary Formation in Core Collapse

At the moment of deepest core collapse, a star cluster core contains less than ten stars. This small number makes the traditional treatment of hard binary formation, assuming a homogeneous background density, suspect. In a previous paper, we have found that indeed the conventional wisdom of binary formation, based on three-body encounters, is incorrect. Here we refine that insight, by further dissecting the subsequent steps leading to hard binary formation. For this purpose, we add some analysis tools in order to make the study less subjective. We find that the conventional treatment does remain valid for direct three-body scattering, but fails for resonant three-body scattering. Especially democratic resonance scattering, which forms an important part of the analytical theory of three-body binary formation, takes too much space and time to be approximated as being isolated, in the context of a cluster core around core collapse. We conclude that, while three-body encounters can be analytically approximated as isolated, subsequent strong perturbations typically occur whenever those encounters give rise to democratic resonances. We present analytical estimates postdicting our numerical results. If we only had been a bit more clever, we could have predicted this qualitative behaviour.

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