Gravothermal Oscillations Research Articles

Gravothermal oscillations in multicomponent models of star clusters

In this paper, gravothermal oscillations are investigated in multicomponent star clusters which have power-law initial mass functions (IMFs). For the power-law IMFs, the minimum masses (mmin) were fixed and three different maximum stellar masses (mmax) 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 multicomponent gas code spedi. The total mass (Mtot) is seen to give an approximate stability condition for power-law IMFs with fixed values of mmax and mmin independent of α. The value Mtot/mmax ≃ 12 000 is shown to give an approximate stability condition which is also independent of mmax, 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.

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.

Effects of Hardness of Primordial Binaries on Evolution of Star Clusters

AbstractWe performed N-body simulations of star clusters with primordial binaries using a new code, GORILLA. It is based on Makino and Aarseth (1992)'s integration scheme on GRAPE, and includes a special treatment for relatively isolated binaries. Using the new code, we investigated effects of hardness of primordial binaries on whole evolution of the clusters. We simulated seven N=16384 equal-mass clusters containing 10% (in mass) primordial binaries whose binding energies are 1, 3, 10, 30, 100, 300, and 1000kT, respectively. Additionally, we also simulated a cluster without primordial binaries and that in which all binaries are replaced by stars with double mass, as references of soft and hard limits, respectively. We found that, in both soft (≤ 3kT) and hard (≥ 1000kT) limits, clusters experiences deep core collapse and shows gravothermal oscillations. On the other hands, in the intermediate hardness (10-300kT), the core collapses halt halfway due an energy releases of the primordial binaries.

Various Modes of Helium Mixing in Globular Cluster Giants and Their Possible Effects on the Horizontal Branch Morphology

It has been known for a long time that some red giants in globular clusters exhibit large star-to-star variations in the abundances of light elements that are not exhibited by field giants. This fact can be taken as evidence that the extra mixing mechanism(s) that operate in globular cluster giants may be consequences of star-star interactions in the dense stellar environment. In order to constrain the extra mixing mechanism(s), we study the influence of helium enrichment along the red giant branch on the evolution of stars through the horizontal branch (HB). Three possible modes of helium enrichment are considered, associated with close encounters of stars in the globular clusters. We show thatasaconsequenceofthevariationsinthecoremass,aswellasinthetotalmassduetomassloss,thecolors of horizontal branch models are distributed over almost the entire range of the horizontal branch. The results are discussedinrelationtoascenariofortheoriginof theabundanceanomaliesandfortheeffectsonthemorphologyof the horizontal branch. We argue that the star-star interactions can not only explain the source of the angular momentum of rapid rotation but also provide a mechanism for the bimodal distribution of rotation rates in some globular clusters. We also propose the time elapsed from the latest core-collapse phase during the gravothermal oscillations as the second parameter to explain the variations in HB morphology among the globular clusters. Subject headings: globularclusters:general — stars:abundances — stars:evolution — stars:horizontal-branch — stars: interiors — stars: rotation

Nonequilibrum behaviour of finite gravitating systems

The behaviour of N equal point masses with an inverse square law of attraction is one of the fundamental problems of statistical physics, because of its numerous applications in astrophysics, and its simplicity. But the simplicity is deceptive. From a theoretical point of view this problem is one of the hardest because it is scale-free, the interaction is long-range, and the interaction exhibits a short-range divergence. Therefore theoretical information is best developed for systems with artificial cutoffs at large and small distances. From the point of view of simulations, the problem is hard because the computational effort grows roughly as N3, and because of fundamental problems in simulating a chaotic system.This talk reviews the relationship between these two approaches, with particular emphasis on simulations of isolated systems (i.e. with no boundary). We emphasise the range of time scales on which different non-equilibrium phenomena operate, and focus on those which are driven by relaxation. We discuss the characteristics of core collapse and gravothermal oscillations, where both basic results of statistical mechanics and phenomenological toy models are particularly instructive. We also review the long-term fate of finite isolated systems.

Star clusters with primordial binaries - I. Dynamical evolution of isolated models

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.

Dynamical simulation of gravothermal catastrophe.

We investigate the dynamical evolution of gravothermal catastrophe in a model of a spherical cluster where, besides the energy and angular momentum, an additional integral of motion is also taken into account. Using dynamical simulation, we study a system of concentric, rotating, spherical shells employing a precise, event-driven, algorithm that permits the controlled exchange of internal angular momentum. Initially the system starts to relax to a locally stable state that is in good agreement with mean field predictions. This is followed by core collapse with the development of a core-halo structure and gravothermal oscillation.

A stochastic Monte Carlo approach to modelling real star cluster evolution -- III. Direct integration of three- and four-body interactions

Spherically symmetric equal mass star clusters containing a large amount of primordial binaries are studied using a hybrid method, consisting of a gas dynamical model for single stars and a Monte Carlo treatment for relaxation of binaries and the setup of close resonant and fly-by encounters of single stars with binaries and binaries with each other (three- and four-body encounters). What differs from our previous work is that each encounter is being integrated using a highly accurate direct few-body integrator which uses regularized variables. Hence we can study the systematic evolution of individual binary orbital parameters (eccentricity, semi-major axis) and differential and total cross sections for hardening, dissolution or merging of binaries (minimum distance) from a sampling of several ten thousands of scattering events as they occur in real cluster evolution including mass segregation of binaries, gravothermal collapse and reexpansion, binary burning phase and ultimately gravothermal oscillations. For the first time we are able to present empirical cross sections for eccentricity variation of binaries in close three- and four-body encounters. It is found that a large fraction of three-body and four-body encounters results in merging. Previous cross sections obtained by Spitzer and Gao for strong encounters can be reproduced, while for weak encounters non-standard processes like formation of hierarchical triples occur.

Monte Carlo Simulations of Globular Cluster Evolution. III. Primordial Binary Interactions

We study the dynamical evolution of globular clusters using our two-dimensional Monte Carlo code with the inclusion of primordial interactions for equal-mass stars. We use approximate analytical cross sections for energy generation from binary-binary and binary-single interactions. After a brief period of slight contraction or expansion of the core over the first few relaxation times, all clusters enter a much longer phase of stable binary burning lasting many tens of relaxation times. The structural parameters of our models during this phase match well those of most observed globular clusters. At the end of this phase, clusters that have survived tidal disruption undergo deep core collapse, followed by gravothermal oscillations. Our results clearly show that the presence of even a small fraction of binaries in a cluster is sufficient to support the core against collapse significantly beyond the normal core-collapse time predicted without the presence of binaries. For tidally truncated systems, collapse is easily delayed sufficiently that the cluster will undergo complete tidal disruption before core collapse. As a first step toward the eventual goal of computing all interactions exactly using dynamical three- and four-body integration, we have incorporated an exact treatment of binary-single interactions in our code. We show that results using analytical cross sections are in good agreement with those using exact three-body integration, even for small fractions, where binary-single interactions are energetically most important.

Multi-Mass gaseous models of globular clusters with stellar evolution

Abstract Gaseous models have proven to be a computationally cheap and reliable tool to investigate the dynamical behaviour of globular clusters. But to compete with other approaches for modelling the evolution of star clusters various important effects have to be taken into account. These are for example the dynamical effects of a stellar mass spectrum, stellar evolution, a consistent treatment of the tidal field of the galaxy, or primordial binaries. Here we report on the effects of a mass spectrum and stellar evolution on the dynamical evolution of the cluster's post-collapse evolution (gravothermal oscillations) and show how only small changes in the mass loss rate can change the internal dynamics of the cluster.

Direct N-body simulations

Special high-accuracy direct force summation N-body algorithms and their relevance for the simulation of the dynamical evolution of star clusters and other gravitating N-body systems in astrophysics are presented, explained and compared with other methods. Other methods means here approximate physical models based on the Fokker–Planck equation as well as other, approximate algorithms to compute the gravitational potential in N-body systems. Questions regarding the parallel implementation of direct “brute force” N-body codes are discussed. The astrophysical application of the models to the theory of relaxing rotating and non-rotating collisional star clusters is presented, briefly mentioning the questions of the validity of the Fokker–Planck approximation, the existence of gravothermal oscillations and of rotation and primordial binaries.

Monte Carlo simulations of star clusters -- I. First Results

A revision of Stodoíkiewicz's Monte Carlo code is used to simulate evolution of star clusters. The new method treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. The first calculations for isolated, equal-mass N-body systems with three-body energy generation according to Spitzer's formulae show good agreement with direct N-body calculations for N = 2000, 4096 and 10 000 particles. The density, velocity, mass distributions, energy generation, number of binaries, etc., follow the N-body results. Only the number of escapers is slightly too high compared with N-body results, and there is no level-off anisotropy for advanced post-collapse evolution of Monte Carlo models as is seen in N-body simulations for N ≤ 2000. For simulations with N > 10 000 gravothermal oscillations are clearly visible. The calculations of N 2000, 4096, 10 000, 32 000 and 100 000 models take about 2, 6, 20, 130 and 2500 h, respectively. The Monte Carlo code is at least 105 times faster than the N-body one for N = 32 768 with special-purpose hardware. Thus it becomes possible to run several different models to improve statistical quality of the data and run individual models with N as large as 100 000. The Monte Carlo scheme can be regarded as a method which lies in the middle between direct N-body and Fokker-Planck models and combines most advantages of both methods.

Monte Carlo simulations of star clusters - I. First Results

A revision of Stodolkiewicz's Monte-Carlo code is used to simulate evolution of star clusters. The new method treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. The first calculations for isolated, equal-mass N-body systems with three-body energy generation according to Spitzer's formulae show good agreement with direct N-body calculations for N=2000, 4096 and 10000 particles. The density, velocity, mass distributions, energy generation, number of binaries etc. follow the N-body results. Only the number of escapers is slightly too high compared to N-body results and there is no level off anisotropy for advanced post-collapse evolution of Monte-Carlo models as is seen in N-body simulations for N 10000 gravothermal oscillations are clearly visible. The calculations of N=2000, 4096, 10000, 32000 and 100000 models take about 2, 6, 20, 130 and 2500 hours, respectively. The Monte-Carlo code is at least 10^5 times faster than the N-body one for N=32768 with special-purpose hardware (Makino 1996ab). Thus it becomes possible to run several different models to improve statistical quality of the data and run individual models with N as large as 100000. The Monte-Carlo scheme can be regarded as a method which lies in the middle between direct N-body and Fokker-Planck models and combines most advantages of both methods.

Two‐Component Fokker‐Planck Models for the Evolution of Isolated Globular Clusters

Two-component (normal and degenerate stars) models are the simplest realization of clusters with a mass spectrum because high-mass stars evolve quickly into degenerates, while low-mass stars remain on the main sequence for the age of the universe. Here we examine the evolution of isolated globular clusters by using two-component Fokker-Planck (FP) models that include heating by binaries formed in tidal capture and in three-body encounters. Three-body binary heating dominates, and the postcollapse expansion is self-similar, at least in models with total mass M ≤ 3 × 105 M☉, initial half-mass radius rh,i ≥ 5 pc, component mass ratio m2/m1 ≥ 2, and number ratio N1/N2 ≤ 300, when m2 = 1.4 M☉. We derive scaling laws for ρc, vi, rc, and rh as functions of m1/m2, N, M, and time t from simple energy-balance arguments; these agree well with the FP simulations. We have studied the conditions under which gravothermal oscillations (GTOs) occur. If Etot and Ec are the energies of the cluster and of the core, respectively, and trh and tc are their relaxation times, then ≡ (Etot/trh)/(Ec/trc) is a good predictor of GTOs: all models with > 0.01 are stable, and all but one with < 0.01 oscillate. We derive a scaling law for against N and m1/m2 and compare them with our numerical results. Clusters with larger m2/m1 or smaller N are stabler.

Mass Segregation and Equipartition of Energy in Two Globular Clusters with Central Density Cusps.

view Abstract Citations (25) References (42) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Mass Segregation and Equipartition of Energy in Two Globular Clusters with Central Density Cusps. Sosin, Craig Abstract We begin by presenting the analysis of a set of deep B- and V-band images of the central density cusp of the globular cluster M30 (NGC 7099), taken with the Faint Object Camera aboard the Hubble Space Telescope. These images are the first to resolve lower-mass main-sequence stars in the cluster's central 10". From the positions of individual stars, we measure an improved position for the cluster center; this new position is 2".6 from the previously known position. We find no evidence of a "flat," constant-surface-density core; however, the data do not rule out the presence of a core of radius up to 1".9 (95% confidence level). We measure a logarithmic cusp slope (d log σ/d log r) of - 0.76±0.07 (1-sigma) for stars with masses between 0.69 and 0.76 Msun, and - 0.82±0.11 for stars with masses between 0.57 and 0.69 Msun. We also compare the overall mass function (MF) of the cluster cusp with the MF of a field at r =4'.6 (near the cluster half-mass radius). The observed degree of mass segregation is well matched by the predictions of an isotropic, multimass King model. We then use the Jeans equation to compare the structure of M30 with that of M15, another cusped cluster, using data from this and a previous paper. We find that M30 is very close to achieving equipartition of energy between stellar species, at least over the observed range in mass and radius, while M15 is not. This difference may be a result of the longer relaxation time in the observed field in M15. The data also suggest that the degree of mass segregation within the two cluster cusps is smaller than one would expect from the measurements at larger radius. If so, this phenomenon might be the result of gravothermal oscillations, of centrally-concentrated populations of binaries, or of a ∼103 Msun black hole in one or more clusters. Publication: The Astronomical Journal Pub Date: October 1997 DOI: 10.1086/118581 arXiv: arXiv:astro-ph/9707251 Bibcode: 1997AJ....114.1517S Keywords: GLOBULAR CLUSTERS: INDIVIDUAL: M30; GLOBULAR CLUSTERS: INDIVIDUAL: M15; GALAXY: STELLAR CONTENT; Astrophysics E-Print: doi:10.1086/118581 full text sources arXiv | ADS | data products SIMBAD (3) MAST (1) ESA (1)

Postcollapse Evolution of Globular Clusters

A number of globular clusters appear to have undergone core collapse, in the sense that their predicted collapse times are much shorter than their current ages. Simulations with gas models and the Fokker-Planck approximation have shown that the central density of a globular cluster after the collapse undergoes nonlinear oscillation with a large amplitude (gravothermal oscillation). However, the question whether such an oscillation actually takes place in real N-body systems has remained unsolved because an N-body simulation with a sufficiently high resolution would have required computing resources of the order of several GFLOPS-yr. In the present paper, we report the results of such a simulation performed on a dedicated special-purpose computer, GRAPE-4. We have simulated the evolution of isolated point-mass systems with up to 32,768 particles. The largest number of particles reported previously is 10,000. We confirm that gravothermal oscillation takes place in an N-body system. The expansion phase shows all the signatures that are considered to be evidence of the gravothermal nature of the oscillation. At the maximum expansion, the core radius is ~1% of the half-mass radius for the run with 32,768 particles. The maximum core size, rc, depends on N as rc ∝ N−1/3.

Fokker-Planck Models of Star Clusters with Anisotropic Velocity Distributions II. Post-Collapse Evolution

The evolution of spherical single-mass star clusters driven by two-body relaxation was followed beyond core collapse by numerically solving the orbit-averaged Fokker-Planck equation in energy–angular momentum space. The heating effect by three-body binaries was incorporated in the Fokker-Planck models. The development of velocity anisotropy after the core collapse is discussed in detail. The anisotropy in the outer regions continues to increase slowly after the collapse. In clusters comprising a relatively small number of stars (N ≲ 10000), the post-collapse expansion is nearly self-similar and the anisotropy at each of inner Lagrangian radii is nearly constant. In clusters comprising a larger number of stars (N ≳ 10000), gravothermal oscillations occur and the anisotropy at each of the inner radii oscillates with the core oscillations. There is no qualitative difference in the nature of the gravothermal oscillations between the isotropic and anisotropic Fokker-Planck models.

Gravothermal Oscillations

We present the first clear evidence that the gravothermal oscillation takes place inN-body systems. We performed directN-body simulations of systems of point-mass particles with particle numbers from 2,048 to 32,768. In the simulation with 32,768 particles, the central density shows an oscillation with an amplitude of ∼ 103, which is similar to what was observed in more approximate models such as a conducting gas sphere and one-dimensional Fokker-Planck calculations. The amplitude is smaller for a smaller number of particles. The number of particles in the core at the maximum contraction is ∼ 10 for all runs, while the number of particles at the maximum expansion is about 0.01N.For 16,384- and 32,768-body runs, the temperature inversion during the expansion phase is clearly visible.

Are Gravothermal Oscillations Gravothermal?

We examine critically the properties of the large-amplitude oscillations seen in Fokker-Planck simulations of globular clusters, with both continuous and stochastic binary heating, and compare them to the defining characteristics of gravothermal oscillations.

Fokker-Planck Models for Rotating Stellar Systems

Observations of Globular Cluster ellipticity distributions related to some fundamental parameters give strong evidence for a decay of rotational energy in these systems with time. In order to study the effectiveness of angular momentum transport (or loss, resp.) a code has been written which solves the Fokker-Planck equation in (E, Jz)-space and follows the evolution from some initial conditions through core collapse (and possibly gravothermal oscillations) up to the post-collapse phase. For the purpose of comparability with N-body simulations rotating initial model configurations according to the prescriptions of Lupton &amp; Gunn (1987) have been constructed. These models are intended to continue previous work by Goodman (1983, Fokker-Planck) and Akiyama &amp; Sugimoto (1989, N-Body). In this contribution the derivation of the flux coefficients is given.