Equal-mass Stars Research Articles

A stochastic Monte Carlo approach to modelling of real star cluster evolution -- I. The model

A new approach is outlined to follow the individual formation and evolution of hard tbree­ body binaries in an evolving equal point mass star cluster. The single stars are treated as a conducting gas sphere with a standard anisotropic gaseous model, binary formation and subsequent hardening by close encounters, and relaxation (dynamical friction) of the binaries with the single stars by a Monte Carlo technique. The results agree very well with the standard theoretical models and with a direct N -body simulation. It is argued that the method can be extended to generate an efficient, cheap and realistic model of large star clusters.

The time-scale for core collapse in spherical star clusters

The collapse time for a cluster of equal-mass stars is usually stated to be either 330 central relaxation times (trc) or 12-19 half-mass relaxation times (trh). But the first of these times applies only to the late stage of core collapse, and the second only to low-concentration clusters. To clarify how the time depends on the density profile, the Fokker-Planck equation is solved for the evolution of a variety of isotropic cluster models, including King models, models with power-law density cusps of ρ ∼ r−γ, and models with nuclei. The collapse times for King models vary considerably with the cluster concentration when expressed in units of trc or trh, but vary much less when expressed in units of trc divided by a dimensionless measure of the temperature gradient in the core. Models with cusps have larger temperature gradients and evolve faster than King models, but not all of them collapse: those with 0 < γ < 2 expand because they start with a temperature inversion. Models with nuclei collapse or expand as the nuclei would in isolation if their central relaxation times are short; otherwise their evolution is more complicated. Suggestions are made for how the results can be applied to globular clusters, galaxies, and clusters of dark objects in the centers of galaxies.Scott D. Tremaine

Dynamics of the Galactic Globular Cluster NGC 3201

B,V CCD frames have been used to derive surface brightness profiles for NGC 3201 out to ~18 arcmin. A total of 857 radial velocities with median precision ~1 km/s for 399 member giants have been used to trace the velocity dispersion profile out to 32' (the approximate tidal radius from fits of single-mass, isotropic King-Michie models to the cluster surface brightness profiles). The median difference in radial velocity for stars on either side of an imaginary axis moved through the cluster in 1 degree steps shows a significant maximum amplitude of 1.22+/-0.25 km/s. We discuss possible explanations of this result, including: (1) cluster rotation; (2) preferential stripping of stars on prograde orbits near the limiting radius; (3) the projection of the cluster space velocity onto the plane of the sky and (4) a slight drift in the velocity zero point. It is difficult to identify the primary cause of the observed velocity field structure unambiguously, and we suspect that all of the above processes may play a role. The B,V surface brightness profiles and radial velocities have been modeled with single- & multi-mass King-Michie models and nonparametric techniques. The density and M/L profiles show good agreement over 1.5<R<10 pc, and both approaches suggest a steady rise in M/L with distance from the cluster center. Due to the low cluster luminosity, we are unable to place useful constraints on the anisotropy of the velocity dispersion profile, though the global mass-to-light ratio is well-constrained by the models as ~2.0 +/-0.2 for the multi-mass and nonparametric models, compared to ~ 1.65 +/-0.15 for models having equal-mass stars. Our best-fit, multi-mass models have mass function slopes of x~0.75 +/-0.25, consistent with findings that mass function depends on the position relative to the potential of the Galaxy.

The Role of Chaos in the Circularization of Tidal Capture Binaries. II. Long-Time Evolution

view Abstract Citations (81) References (28) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Role of Chaos in the Circularization of Tidal Capture Binaries. II. Long-Time Evolution Mardling, Rosemary A. Abstract We examine the long-time dynamical evolution of binaries formed by tidal capture using a self-consistent normal mode analysis. An efficient method for handling very long period orbits is developed which allows us to examine the highly eccentric phase following capture. We start with an adiabatic analysis and show that the orbital eccentricity performs a random walk between the initial eccentricity (≥ 1) and some nonzero minimum value. The existence of a minimum eccentricity in turn restricts the maximum tidal energy the system is capable of acquiring. We then examine dissipative systems by introducing an artificial mode damping mechanism and show that tidal capture binaries go through two distinct phases. Following capture is a short violent chaotic phase during which the dynamical behavior is as described by the adiabatic analysis. Huge tides are raised, and the tidal luminosity is likely to be large if nonlinear dissipative processes are efficient. Given that the binary survives this phase, it must dissipate a prescribed amount of energy before it enters the second phase during which the orbital behavior is no longer chaotic and the orbital eccentricity varies quasi-periodically. During this long quiescent phase, the tides are small and the binary circularizes only via dissipative processes. The prescribed energy loss and the eccentricity at the beginning of this phase depend on the periastron separation at capture. For example, for equal mass stars with an initial periastron separation of 3 stellar radii, the system starts this phase with an eccentricity of 0.8. For such a system, the quiescent circularization phase is estimated to be at least 107 yr, in contrast to a circularization time predicted by the standard model of 10 yr (McMillan, McDermott, & Taam 1987). Since capture orbits are chaotic, there exists the possibility of a binary returning to an unbound state. We show that the majority of tidal capture binaries are stable against self-ionization. We discuss the various destructive processes which the stars must be stable against during the chaotic phase in order to enter the quiescent phase intact. We also discuss the implications for the formation of low-mass X-ray binaries and related objects and the role of tidal capture binaries in globular cluster evolution. Publication: The Astrophysical Journal Pub Date: September 1995 DOI: 10.1086/176179 Bibcode: 1995ApJ...450..732M Keywords: CELESTIAL MECHANICS; STELLAR DYNAMICS; CHAOS; GALAXY: GLOBULAR CLUSTERS: GENERAL; STARS: BINARIES: CLOSE; STARS: PULSARS: INDIVIDUAL ALPHANUMERIC: PSR 2127+11C; X-RAYS: STARS full text sources ADS | data products SIMBAD (2) Related Materials (1) Part 1: 1995ApJ...450..722M

Influence of the stellar mass function of the evaporation rate of tidally limited postcollapse globular clusters

view Abstract Citations (47) References (50) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Influence of the Stellar Mass Function on the Evaporation Rate of Tidally Limited Postcollapse Globular Clusters Lee, Hyung Mok ; Goodman, Jeremy Abstract We study the rate of escape of stars (evaporation) from tidally limited postcollapse globular clusters having a power-law distribution of stellar masses. We use a multimass Fokker-Planck code and assume a steady tidal field. Stellar-dynamical processes cause the inner parts of the cluster to expand, which in turn causes stars to overflow the tidal boundary. Mass loss by stellar evolution is assumed to be unimportant in these later evolutionary stages. The fraction of the cluster mass lost per half-mass relaxation time trh is roughly constant, in agreement with simple homologous models with equal-mass stars. If trh is computed in the conventional way from the mean stellar mass, however, a broad stellar mass function can double the loss of mass per trh. We discuss implications of our results for the evolution of globular cluster systems in our own and other galaxies. In particular, the number of Galactic clusters destroyed by evaporation alone may be as large as or larger than the present population. Publication: The Astrophysical Journal Pub Date: April 1995 DOI: 10.1086/175506 Bibcode: 1995ApJ...443..109L Keywords: Astronomical Models; Evaporation Rate; Globular Clusters; Stellar Evolution; Stellar Mass; Stellar Orbits; Celestial Mechanics; Galactic Evolution; Milky Way Galaxy; Relaxation Time; Star Distribution; Astrophysics; CELESTIAL MECHANICS; STELLAR DYNAMICS; GALAXY: GLOBULAR CLUSTERS: GENERAL full text sources ADS |

The orbit of the double-line spectroscopic binary system HR 7112

We present velocity observations, obtained with the E. W. Fick Observatory 0.6 m telescope, of the 7th magnitudeK1 star HR 7112. Our observations show that HR 7112 is a double-line spectroscopic binary system composed of almost equal mass stars with a period of 215.6 days. The orbit has an eccentricitye = 0.142 and the stars orbit each other with a mean semi-major axis distance of approximately 0.4sin(i) AU. The masses of the stars are consistent with the classification as giant stars. Tidal effects are known to circularise the orbits of giant stars and this process is apparently underway in this system.

Globular cluster photometry with the Hubble Space Telescope. 2: U, V, and I measurements of M15

The projected density distribution of resolved stars near the center of M15 is shown to be consistent with either a power-law cusp N(r) approximately r(exp alpha), with alpha approximately -0.85 +/- 0.2, or with a King model with a core of radius approximately less than 2 sec. The inferred slope is in agreement with the theoretical value, alpha = -0.75, calculated by Bahcall and Wolf for the distribution of equal-mass stars surrounding a massive black hole and is also consistent with the radial profile expected from core collapse without a central black hole. The object AC 214 is a candidate for the central density cusp. Analysis of Monte Carlo simulations of the diffuse light indicates that, using current analysis techniques and available data, the residual light is not a reliable indicator of the true density distribution. This is contrary to earlier work. Photometric measurements in V and I of more than 5 x 10(exp 3) stars (and in U, V, and I of approximately greater than 1500 stars) are used to construct color-magnitude diagrams in the central 1 min of M15. Fourteen blue straggler candidates are identified in the inner 20 sec. The central color gradient noticed by previous researchers is caused by a central depletion of bright red giant stars rather than an excess of blue stragglers or blue horizontal branch stars.

The onset of gravothermal oscillations in globular cluster evolution

We have carried out an extensive set of Fokker-Planck simulations of the evolution of globular clusters on very long timescales, up to 600 times the initial core collapse time t(sub cc). We consider an idealized equal mass star cluster, with a wide range of values for the total number of stars, 7000 less than N less than 2 x 10(exp 6). Our models include the heating effect of compact binaries formed in three-body encounters, which halts the initial core collapse and drives a core reexpansion. Postcollapse gravothermal oscillations of the cluster core are found to occur for all N approximately greater than or equal to 8000. For 8000 approximately less than or equal to N approximately less than or equal to 11,000, the oscillation has a simple, regular waveform with a single, well-defined period. For N approximately equals 12,000, the oscillations become nonlinear in a process resembling a period doubling. For N approximately greater than or equal to 14,000, the waveform of the oscillations becomes increasingly more irregular with increasing N, resembling chaotic behavior for N approximately greater than or equal to 15,000. During the oscillations, the core radius and core mass vary dramatically: by more than a factor of 10 for N greater than 15,000, by more than a factor of 100 for N greater than 5 x 10(exp 4), and by more than a factor of 1000 for N greater than 5 x 10(exp 5). However, even during the times of maximum expansion, the core contains only a small fraction of the cluster mass. For most N values, the maximum core mass at any time after core collapse is less than 1% of the cluster mass. The exceptions lie in the range 5 x 10(exp 4) approximately less than or equal to N approximately equal to or less than 2 x 10(exp 5), where the maximum post-collapse core mass reaches approximately 2% of the cluster mass. We discuss the observational implications of these predictions.

ENCOUNTERS WITH PROTOSTELLAR DISKS

ABSTRACT A numerical study of encounters between stars with circumstellar disks has bee completed. Cross sections and rates for disk tilt, disk disruption, and binary formation are estimated using a large data base of encounter simulations. The consequences of these results for star-forming regions and our solar system are discussed. A numerical code is developed which is capable of evolving a mixture of stars and gas in three dimensions. The algorithm is based on the method of smoothed-particle hydrodynamics combined with the heirarchical tree method of computing gravitational forces. The code is tested by simulating the collision between two sheets of gas and the radial pulsations of a polytropic gas sphere. A protostellar-disk model is developed based on simple assumptions. Test encounters are performed to determine the sensitivity of measured quantities on algorithm parameters, such as the gravitational tolerance and viscosity. It is shown that the solar system could have had an encounter shortly after its formation of sufficient strength to generate the observed obliquity yet retain enough mass and radial extent to form the planetary system. For the Orion B clusters as a whole, it is estimated that during a one-million-year period of time a few percent of the stars will experience an enoucnter that results in a disk tilt of 7 degrees or greater. For the central regions of NGC 2024 and the Trapezium cluster values of 24% and 39% are obtained, respectively. Encounters between equal-mass stars with periastra of 0.5, 1.0, 1.5, and 2.0 disk radii will retain on average about 15%, 40%, 55%, and 75% of the disk mass, respectively. For encounters that do not penetrate the disk a minimum of 15% of the mass is retained. Even in dense environments the characteristic lifetime of a disk due to disruptive encounters can be many millions of years. On average, an encounter that penetrates the disk will dissipate an amount of orbital energy equal to approximately 50% of the initial binding energy of the disk. For the central region of NGC 2024 a star-disk capture rate is estimated at Gammac=0.03 and 0.07 Myr-1 for disk masses of 0.1 and 0.5 solar mass, respectively. The equivalent disk-disk capture rates are ≥0.05 and 0.08 Myr-1. The most favorable rate implies that during a 1-Myr period of time 16% of the stars will be captured into a binary. These rates are lower limits because of factors such as higher stellar densities in the past due to cluster expansion, the presence of substructure, and the existence of binary and higher-order systems. In particular, encounters may be a natural outcome of the star-formation process, at a time when protostars are surrounded by large massive disks and extended envelopes of material.

Coalescence of Spinning Binary Neutron Stars of Equal Mass

We have performed numerical simulations of coalescence of binary neutron stars using a Newtonian hydrodynamics code including radiation reaction by gravitational waves. In order to examine the effect of spin, we start the simulations from two distinct types of the initial conditions. In the first case, to see the dependence of results on the initial separation of binary, a Roche solution of separation 27 km, each mass ∼1.5 M⊙ and each radius ∼9 km, respectively, is given as the initial condition. We found that the evolution sequence and the wave form of gravitational waves are essentially the same as those in the previous simulations in which computations are started when two neutron stars just contact. In the second case, we include spin of each star with -Ω, where Ω is Keplerian angular velocity of orbital motion, which is required from the conservation of the circulation. We found that the wave pattern has high amplitude oscillation after coalescence contrary to the former case. This means that it will be possible to determine the spin of coalescing neutron stars from the observed wave form of gravitational waves. The maximum amplitude of gravitational waves is 3.4 ×10-21 for a hypothetical event at the distance of 10 Mpc.

Star cluster evolution with primordial binaries. I - A comparative study

view Abstract Citations (102) References (51) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Star Cluster Evolution with Primordial Binaries. I. A Comparative Study McMillan, Steve ; Hut, Piet ; Makino, Junichiro Abstract The evolution of equal-mass star clusters containing a mass fraction of about 20 percent binaries has been followed using direct integration, making one run each for a total number of stars of N = 282 and N = 563, and four runs for N = 1126. For comparison the evolution of an equivalent star system where the binaries were replaced by stars twice as heavy as the other stars was followed. The pre-core-collapse evolution is driven by mass segregation between the equal-mass single stars and the binaries, which are twice as heavy. After core collapse, the cluster shows, on average, a smooth reexpansion driven by a steady rate of burning (hardening) of primordial binaries. With so much primordial fuel present, the postcollapse cluster core is significantly larger than is the case in comparison runs without primordial binaries. Publication: The Astrophysical Journal Pub Date: October 1990 DOI: 10.1086/169289 Bibcode: 1990ApJ...362..522M Keywords: Binary Stars; Star Clusters; Stellar Evolution; Stellar Mass; Computer Programs; Computerized Simulation; Globular Clusters; Hardware; Star Distribution; Astrophysics; CLUSTERS: OPEN; STARS: BINARIES; STARS: EVOLUTION; STARS: STELLAR DYNAMICS full text sources ADS | Related Materials (2) Part 2: 1991ApJ...372..111M Part 3: 1994ApJ...427..793M

Three-dimensional hydrodynamical simulations of stellar collisions. I - Equal-mass main-sequence stars

view Abstract Citations (196) References (20) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Three-dimensional Hydrodynamical Simulations of Stellar Collisions. I. Equal-Mass Main-Sequence Stars Benz, Willy ; Hills, Jack G. Abstract Two distinct mass-loss mechanisms are noted in the present, fully three-dimensional calculations of collisions between identical stars. While strong shocks in nearly head-on collisions lead to high-velocity jets perpendicular to the collision axis, with increasing mass loss as impact velocity at infinity increases from zero to 2.3 times the escape velocity from the stellar surface, low velocity encounters lead to a sharp increase in mass loss at impact parameters that correspond to nearly-grazing collisions in a two-stage process. In the first stage, the two stars become gravitationally bound due to the encounter's energy dissipation; these binary components then violently coalesce during subsequent periastron passage. Publication: The Astrophysical Journal Pub Date: December 1987 DOI: 10.1086/165857 Bibcode: 1987ApJ...323..614B Keywords: Collisions; Computational Astrophysics; Hydrodynamic Equations; Stellar Motions; Stellar Physics; Binary Stars; Computerized Simulation; Galactic Nuclei; Stellar Mass Ejection; Astrophysics; HYDRODYNAMICS; NUMERICAL METHODS; STARS: BINARIES; STARS: MASS LOSS; STARS: STELLAR DYNAMICS full text sources ADS | Related Materials (2) Part 2: 1989ApJ...342..986B Part 3: 1992ApJ...389..546B

ORBIT AND PARALLAX OF SIGMA 1819.

ABSTRACT The paper presents a new parallax, proper motion, and orbit for the 7th magnitude G0 V pair Sigma 1819. The absolute parallax is +0.027 + or - 0.006 arcsec (s.e.). With the newly determined orbital elements (P = 220.0 y, a = 1.117 arcsec) the components are found to be main-sequence stars of equal mass (about 0.74 the solar mass) with absolute magnitudes of 4.7 and 4.8.

Close Encounters

The mathematical treatment of encounters is discussed briefly, and it is pointed out that in a spherical system with an isotropic distribution function and equal-mass stars, the relevant equations can be analytically orbit-averaged. Even in small systems, close encounters have little effect on core collapse. Near a black hole, however, close encounters may cause an accumulation of stars on large, very eccentric orbits.

Evolution of star clusters with a mass distribution of binaries

The dynamical evolution of star clusters with a initial population of binaries strongly depends on the parameters of the bounded pairs. We performed computational simulation with directN-body codes, on models withN=300 objects, with initial fraction of 20% of binaries with the same orbital semi-axis and with a Salpeter mass spectrum. The results are compared with previous analysis on similar equal-mass star models. The calculation points out that binaries with different mass companions determine an enhanced and sudden expansion of the system with neither core collapse nor central mass segregation.

Encounters of binaries – II. Unequal energies

Gravitational encounters of pairs of hard binaries with unequal energies, but equal mass stars, have been studied numerically. Special attention was drawn to the phenomenon called collision, which cannot be treated as a fly-by. A semi-analytical theory for obtaining the collision cross-section and outcome distributions was developed and the free parameters fitted to numerical observations. Good agreement between theory and experiments confirms the theoretical assumption that the class of interactions, which results in a disruption into one binary and two separately escaping stars, can be treated in terms of two independent random ejections. The energy distribution of the first ejection is approximately scaled by the harmonic mean of the binding energies of the binaries as corrected somewhat for the impact energy, while the disruption of the remaining three-body system has its total energy as the scaling factor. With increasing binding energy ratio the number of hierarchical three-body systems, resulting from collisions, becomes larger. For equal energies it is 20 per cent and gets the value 50 per cent if the binding energies differ by a factor of about 4.

On formation of close binaries by two-body tidal capture

We calculate in detail the two-body tidal capture mechanism of Fabian, Pringle, and Rees: when two unbound stars have a close encounter, they may become bound by the energy that each deposits into nonradial oscillations of the other. After dimensional scalings are removed, the process depends only on a single dimensionless parameter, and on the dimensionless envelope structure of the stars. General formulae are derived; for definiteness, we apply them to the specific case of stars with an n=3 polytropic structure. Capture cross sections as a function of velocity and capture rates for an isothermal distribution are given for the case of equal-mass stars; other cases can easily be computed from the formulae given.

Steepest-descent technique and stellar equilibrium statistical mechanics. I - Newtonian clusters in a box

A microcanonical statistical mechanics formulation has been developed for nonrotating systems of stars of equal mass. The system, placed in a confining volume and with a cutoff of the interparticle gravitational interaction at short distances, can have thermodynamic equilibrium states. Sequences of equilibrium states are presumed to simulate slowly evolving, near-equilibrium configurations of real star clusters. An exact functional expression for the entropy of such systems is derived which has also a relativistic counterpart. The entropy is evaluated in an approximation which is mean field plus fluctuations. Evaluations beyond this approximation can readily be carried out. We obtain the necessary and sufficient conditions for spherically symmetric clusters to be thermodynamically stable about a mean field solution, with respect to arbitrary fluctuations in the microcanonical ensemble. The stability conditions amount to the following quantities having definite signs: (i) a functional form, quadratic in ''mean field'' fluctuations, (ii) the derivative of the gravito-chemical potential with respect to the number of particles, at fixed temperature, being positive definite, and (iii) the heat capacity C/sub ..nu../, at fixed number of particles, being positive definite. In a sequence of equilibrium configurations in which the ratio of densities between the center and the boundary of themore » cluster is progressively increased, conditions (i) and (ii) break down simultaneously when this density contrast is equal to 1.58. Condition (i) remains unsatisfied for higher density contrasts. The limit 1.58 on the density contrast is much more stringent than that given by condition (iii) which breaks down only for a value of 32.1. Our results are in sharp contrast to those of Antonov's criterion according to which instabilities appear when the density contrast is higher than 709. Time scales of evolutions of various unstable configurations are not considered in this work. (AIP)« less

The Double-Lined Binary Alpha Octantis.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Double-Lined Binary Alpha Octantis. Buscombe, W. ; Morris, P. M. Abstract Radial velocities have been oneasured from 24 new spectrograms of HD 199532, which together with eight measures previously published from Lick Observatory, make possible a solution of the orbit. The dispersion of both sets of plates is about 36 A/mm at H~. On six of the Mount Stromlo spectrograms, double absorption lines are resolved. The measures indicate velocity curves of the same amplitude (47 km/sec) corresponding to stars of equal mass. The period is 9.073 days, and the system velocity +45 km/sec. The MK classes are F4 III+F5 III. The eccentricity is 0.39 and primary periastron in longitude 276~13' at JD 2435302.404. The mass-function 0.304 combined with masses each twice the sun's, corresponding to the luminosities, would indicate an inclination 23 between the normal to the plane of the orbit and our line of sight. As the projected semi-major axis is 5.4X 100 km, and the stellar radii are 3X 1~ km, only very shallow grazing eclipses may be possible. Publication: The Astronomical Journal Pub Date: 1960 DOI: 10.1086/108096 Bibcode: 1960AJ.....65R..50B full text sources ADS |

Self-gravitating Star Systems

The Boltzmann equation for a star system in which the controlling forces are solely due to the gravitational attractions between the members is not strictly linear. For the restricted case of a system of stars of equal mass, stratified in plane parallel layers, the Boltzmann equation reduces to a nonlinear partial differential equation in three independent variables. Three functions which satisfy this equation have been found. Two of these correspond to the equilibrium configurations of an isothermal self-gravitating gas, and of an adiabatic gas. The hypothetical models can be compared with the greatly flattened galactic system, and there is considerable agreement between the observed variation of density with distance from the galactic plane and the variation predicted by one of the three solutions.