Results Of N-body Simulations Research Articles

Description of quasi-equilibrium states of self-gravitating systems based on non-extensive thermostatistics

We present an application of Tsallis’ generalized entropy to analysis for thermodynamical nature of the self-gravitating stellar systems. Extremizing the Tsallis entropy, we obtain an equation of the state known as stellar polytrope. And a role of the extremes as a quasi-equilibrium is shown from results of N-body simulations.

Self-gravitating stellar systems and non-extensive thermostatistics

After introducing the fundamental properties of self-gravitating systems, we present an application of Tsallis' generalized entropy to the analysis of their thermodynamic nature. By extremizing the Tsallis entropy, we obtain an equation of state known as the stellar polytrope. For a self-gravitating stellar system confined within a perfectly reflecting wall, we discuss the thermodynamic instability caused by its negative specific heat. The role of the extremum as a quasi-equilibrium is also demonstrated from the results of N-body simulations.

FORMATION AND EVOLUTION OF SELF-INTERACTING DARK MATTER HALOS

Observations of dark matter dominated dwarf and low surface brightness disk galaxies favor density profiles with a flat-density core, while cold dark matter (CDM) N-body simulations form halos with central cusps, instead. This apparent discrepancy has motivated a re-examination of the microscopic nature of the dark matter in order to explain the observed halo profiles, including the suggestion that CDM has a non-gravitational self-interaction. We study the formation and evolution of self-interacting dark matter (SIDM) halos. We find analytical, fully cosmological similarity solutions for their dynamics, which take proper account of the collisional interaction of SIDM particles, based on a fluid approximation derived from the Boltzmann equation. The SIDM particles scatter each other elastically, which results in an effective thermal conductivity that heats the halo core and flattens its density profile. These similarity solutions are relevant to galactic and cluster halo formation in the CDM model. We assume that the local density maximum which serves as the progenitor of the halo has an initial mass profile <TEX>${\delta}M / M {\propto} M^{-{\epsilon}$</TEX>, as in the familiar secondary infall model. If <TEX>$\epsilon$</TEX> = 1/6, SIDM halos will evolve self-similarly, with a cold, supersonic infall which is terminated by a strong accretion shock. Different solutions arise for different values of the dimensionless collisionality parameter, <TEX>$Q {\equiv}{\sigma}p_br_s$</TEX>, where <TEX>$\sigma$</TEX> is the SIDM particle scattering cross section per unit mass, <TEX>$p_b$</TEX> is the cosmic mean density, and <TEX>$r_s$</TEX> is the shock radius. For all these solutions, a flat-density, isothermal core is present which grows in size as a fixed fraction of <TEX>$r_s$</TEX>. We find two different regimes for these solutions: 1) for <TEX>$Q < Q_{th}({\simeq} 7.35{\times} 10^{-4}$</TEX>), the core density decreases and core size increases as Q increases; 2) for <TEX>$Q > Q_{th}$</TEX>, the core density increases and core size decreases as Q increases. Our similarity solutions are in good agreement with previous results of N-body simulation of SIDM halos, which correspond to the low-Q regime, for which SIDM halo profiles match the observed galactic rotation curves if <TEX>$Q {\~} [8.4 {\times}10^{-4} - 4.9 {\times} 10^{-2}]Q_{th}$</TEX>, or <TEX>${\sigma}{\~} [0.56 - 5.6] cm^2g{-1}$</TEX>. These similarity solutions also show that, as <TEX>$Q {\to}{\infty}$</TEX>, the central density acquires a singular profile, in agreement with some earlier simulation results which approximated the effects of SIDM collisionality by considering an ordinary fluid without conductivity, i.e. the limit of mean free path <TEX>${\lambda}_{mfp}{\to} 0$</TEX>. The intermediate regime where <TEX>$Q {\~} [18.6 - 231]Q_{th}$</TEX> or <TEX>${\sigma}{\~} [1.2{\times}10^4 - 2.7{\times}10^4] cm^2g{-1}$</TEX>, for which we find flat-density cores comparable to those of the low-Q solutions preferred to make SIDM halos match halo observations, has not previously been identified. Further study of this regime is warranted.

Negative skewness of radial pairwise velocity in the quasi-non-linear regime: Zel'dovich approximation

According to N-body numerical simulations, the radial pairwise velocities of galaxies have negative skewness in the quasi-non-linear regime. To understand its origin, we calculate the probability distribution function of the radial pairwise velocity using the Zel'dovich approximation, i.e. an analytical approximation for gravitational clustering. The calculated probability distribution function is in good agreement with the result of N-body simulations. Thus, the negative skewness originates in relative motions of galaxies in the clustering process that the infall dominates over the expansion.

A unified theory of the formation of galactic bars

We give arguments for a basically unified formation mechanism of slow (Lynden-Bell) and fast (common) galactic bars. This mechanism is based on an instability that is akin to the well-known instability of radial orbits and is produced by the mutual attraction and alignment of precessing stellar orbits (so far, only the formation of slow bars has been explained in this way). We present a general theory of the low-frequency modes in a disk that consists of orbits precessing at different angular velocities. The problem of determining these modes is reduced to integral equations of moderately complex structure. The characteristic pattern angular velocities Ωp of the low-frequency modes are of the order of the mean orbital precession angular velocity \(\bar \Omega _{pr}\). Bar modes are also among the low-frequency modes; while \(\Omega _p \approx \bar \Omega _{pr}\) for slow bars, Ωp for fast bars can appreciably exceed even the maximum orbital precession angular velocity in the disk Ωprmax (however, it remains of the order of these precession angular velocities). The possibility of such an excess of Ωp over Ωprmax is associated with the effect of “repelling” orbits. The latter tend to move in a direction opposite to the direction in which they are pushed. We analyze the pattern of orbital precession in potentials typical of galactic disks. We note that the maximum radius of an “attracting” circular orbit rc can serve as a reasonable estimate of the bar length lb. Such an estimate is in good agreement with the available results of N-body simulations.

Evolution of star clusters moving on elliptic orbits

We describe results of N-body simulations of star clusters moving on elliptic orbits. Our aim is to follow the evolution of these clusters and study how their dissolution times depend on particle number and eccentricity.We find that the scaling of the lifetimes is nearly independent of the eccentricity, despite the fact that the amount of tidal shocking changes with eccentricity. In addition, no leveling off of the lifetimes at the high-N end, which would indicate the influence of tidal shocking, is seen.

Density profiles of dark matter halos with anisotropic velocity tensors

We present density profiles, that are solutions of the spherical Jeans equation, derived under the following two assumptions: (i) the coarse grained phase-density follows a power-law of radius, ρ/σ 3 ∝ r −α , and (ii) the velocity anisotropy parameter is given by the relation β a ( r)= β 1+2 β 2 ( r/ r *)/[1+( r/ r *) 2] where β 1, β 2 are parameters and r * equals twice the virial radius, r vir, of the system. These assumptions are well motivated by the results of N-body simulations. Density profiles have increasing logarithmic slopes γ, defined by γ=− d ln ρ d ln r . The values of γ at r=10 −2.5 r vir, a distance where the systems could be resolved by large N-body simulations, lie in the range 1.0–1.6. These inner values of γ increase for increasing β 1 and for increasing concentration of the system. On the other hand, slopes at r= r vir lie in the range 2.42–3.82. A model density profile that fits well the results at radial distances between 10 −3 r vir and r vir and connects kinematic and structural characteristics of spherical systems is described.

Long-term evolution of isolated N-body systems

We report results of N-body simulations of isolated star clusters, performed up to the point where the clusters are nearly completely dissolved. Our main focus is on the post-collapse evolution of these clusters. We find that, after core collapse, isolated clusters evolve along nearly a single sequence of models, the properties of which are independent of the initial density profile and particle number. Because of the slower expansion of high-N clusters, relaxation times become almost independent of the particle number after several core collapse times, at least for the particle range of our study. As a result, the dissolution times of isolated clusters exhibit a surprisingly weak dependence on N. We find that most stars escape as a result of encounters between single stars inside the half-mass radius of the cluster. Encounters with binaries take place mostly in the cluster core and account for roughly 15 per cent of all escapers. Encounters between single stars at intermediate radii are also responsible for the build-up of a radial anisotropic velocity distribution in the halo. For clusters undergoing core oscillations, escape owing to binary stars is efficient only when the cluster centre is in a contracted phase. Our simulations show that it takes about 105N-body time-units until the global anisotropy reaches its maximum value. The anisotropy increases with particle number, and it seems conceivable that isolated star clusters become vulnerable to radial orbit instabilities for large enough N. However, no indication for the onset of such instabilities was seen in our runs.

B1422+231: The influence of mass substructure on strong lensing

In this work we investigate the gravitationally lensed system B1422+231. High-quality VLBI image positions, fluxes and shapes as well as an optical HST lens galaxy position are used. First, two simple and smooth models for the lens galaxy are applied to t observed image positions and fluxes; no even remotely acceptable model was found. Such models also do not accurately reproduce the image shapes. In order to t the data successfully, mass substructure has to be added to the lens, and its level is estimated. To explore expectations about the level of substructure in galaxies and its influence on strong lensing, N-body simulation results of a model galaxy are employed. By using the mass distribution of this model galaxy as a lens, synthetic data sets of dierent four image system congurations are generated and simple lens models are again applied to t them. The diculties in tting these lens systems turn out to be similar to the case of some real gravitationally lensed systems, thus possibly providing evidence for the presence and strong influence of substructure in the primary lens galaxy.

Evolution of Planetesimal Velocities Based on Three-Body Orbital Integrations and Growth of Protoplanets

We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus 137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus 106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10–30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies.

Angular momentum induced phase transition in spherical gravitational systems

We present the results of N-body simulations for a self-gravitating system in the microcanonical ensemble for the l 2 model, where we fix the magnitude of angular momentum for each particle. In the mean-field limit, the system exhibits a first-order phase transition between a centrally condensed and a quasi-uniform phase in both canonical and microcanonical ensembles. In the present work, we verify the presence of the phase transition via N-body simulations. With increasing particle numbers, the simulation results converge to the mean-field phase transition in accordance with finite size scaling theory. We also use correlations in position and time to investigate the main dynamical features of each phase.

Testing the modified Press-Schechter model against N-body simulations

A modified version of the extended Press–Schechter model for the growth of dark-matter haloes was introduced in two previous papers, with the aim of explaining the mass–density relation shown by haloes in high-resolution cosmological simulations. In this model, major mergers are well separated from accretion, thereby allowing a natural definition of halo formation and destruction. This makes it possible to derive analytic expressions for halo formation and destruction rates, the mass accretion rate and the probability distribution functions of halo formation times and progenitor masses. The stochastic merger histories of haloes can be readily derived and easily incorporated into semi-analytical models of galaxy formation, thus avoiding the usual problems encountered in the construction of Monte Carlo merger trees from the original extended Press–Schechter formalism. Here we show that the predictions of the modified Press–Schechter model are in good agreement with the results of N-body simulations for several scale-free cosmologies.

Is the Fast Evolution Scenario for Virialized Compact Groups Really Compelling? The Role of a Dark Massive Group Halo

We report on results of N-body simulations aimed at testing the hypothesis that galaxies in X-ray-emitting (i.e., virialized) compact groups are not tidally stripped when they are embedded in a common, massive, quiescent dark matter halo. To disentangle the effects of interactions from spurious effects due to an incorrect choice of the initial galaxy model configurations, these have been chosen to be tidally limited King spheres, representing systems in quasi-equilibrium within the tidal field of the halo. The potential of the halo has been assumed to be frozen and the braking due to dynamical friction neglected. Our results confirm the hypothesis of low rates of tidal stripping and suggest a scenario for virialized compact group evolution in their quiescent phases where only very moderate tidally induced galaxy evolution can generally be expected. This implies the group is stable, provided that the dynamical friction timescales in these systems are not much shorter than the Hubble time. We discuss briefly this possibility, in particular taking account of the similarity between the velocity dispersions of a typical virialized compact group and the internal velocity dispersion of typical member galaxies.

Analytic model for galaxy and dark matter clustering

We investigate an analytic model to compute nonlinear power spectrum of dark matter, galaxies and their cross-correlation. The model is based on Press-Schechter halos, which cluster and have realistic dark matter profiles. The total power spectrum is a sum of two contributions, one from correlations betwen the halos and one from correlations within the same halo. We show that such a model can give dark matter power spectra which match well with the results of N-body simulations, provided that concentration parameter decreases with the halo mass. Galaxy power spectrum differs from dark matter power spectrum because pair weighted number of galaxies increases less rapidly than the halo mass, as predicted by theoretical models and observed in clusters. In this case the resulting power spectrum becomes a power law with the slope closed to the observed. Such a model also predicts a later onset of nonlinear clustering compared to the dark matter, which is needed to reconcile the CDM models with the data. Generic prediction of this model is that bias is scale dependent and nonmonotonic. For red or elliptical galaxies bias in power spectrum may be scale dependent even on very large scales. Our predictions for galaxy-dark matter correlations, which can be observed through the galaxy-galaxy lensing, show that these cannot be interpreted simply as an average halo profile of a typical galaxy, because different halo masses dominate at different scales and because larger halos host more than one galaxy. We discuss the prospects of using cross-correlations in combination with galaxy clustering to determine the dark matter power spectrum (ABRIDGED).

Cosmological Redshift‐Space Distortion on Clustering of High‐Redshift Objects: Correction for Nonlinear Effects in the Power Spectrum and Tests withN‐Body Simulations

We examine the cosmological redshift-space distortion effect on the power spectrum of objects at high redshifts,which is an unavoidable observational contamination in general relativistic cosmology. In particular, we consider the nonlinear effects of density and velocity evolution using high-resolution N-body simulations in cold dark matter models. We find that theoretical modeling on the basis of the fitting formulae of nonlinear density and velocity fields accurately describes the numerical results, especially in quasi-nonlinear regimes. These corrections for nonlinear effects are essential in order to use the cosmological redshift-space distortion as a cosmological test. We perform a feasibility test to derive constraints from the future catalogs of high-redshift quasars using our theoretical modeling and results of N-body simulations. Applying the present methodology to the future data from ongoing surveys of high-redshift galaxies and quasars will provide a useful tool to constrain a geometry of the universe.

N-body simulations of interactions and mergings in small galaxy groups

AbstractIn this paper I focus on three topics related to the dynamical evolution of small galaxy groups, for which the input of N-body simulations has been decisive. These are the merging rates in compact groups, the properties of remnants of multiple mergers, and the evolution of disc galaxies surrounded by one or more satellites. The short dynamical times of compact groups make it difficult to understand why such groups are observed at all. N-body simulations have pointed out two possible classes of solutions to this problem. The first one proposes that there is ongoing formation of compact groups, or that the longevity of the group is due to secondary infall. For the second class of solutions the longevity of compact groups is due either to their specific initial conditions, or to a massive common halo, encompassing the whole group. I discuss here these alternatives, together with their respective advantages and disadvantages. I then turn to the structure of remnants of multiple mergers and compare the results of N-body simulations with the properties of observed elliptical galaxies. Finally I discuss the dynamical evolution of a disc galaxy surrounded by one or more spherical satellites.

Predictions for the clustering properties of the Lyman-alpha forest -- I. One-point statistics

We present predictions for the one-point probability distribution and cumulants of the transmitted QSO flux in the high redshift Lyman-$\alpha$ Forest. We make use of the correlation between the Lyman-$\alpha$ optical depth and the underlying matter density predicted by gravitational instability theory and seen in numerical hydrodynamic simulations. We have modelled the growth of matter fluctuations using the non-linear shear-free dynamics, an approximation which reproduces well the results of perturbation theory for the cumulants in the linear and weakly non-linear clustering regime. As high matter overdensities tend to saturate in spectra, the statistics of the flux distribution are dominated by weakly non-linear overdensities. As a result, our analytic approach can produce accurate predictions, when tested against N-body simulation results, even when the underlying matter field has rms fluctuations larger than unity. Our treatment can be applied to either Gaussian or non-Gaussian initial conditions. Here we concentrate on the former case, but also include a study of a specific non-Gaussian model. We discuss how the methods and predictions we present can be used as a tool to study the generic clustering properties of the \lya forest at high-redshift. With such an approach, rather than concentrating on simulating specific cosmological models, we may be in the position to directly test our assumptions for the Gaussian nature of the initial conditions, and the gravitational instability origin of structure itself. In a separate paper we present results for two-point statistics.

Non-linearity and stochasticity in the density-velocity relation

We present results of the investigations of the statistical properties of a joint density and velocity divergence probability distribution function (PDF) in the mildly non-linear regime. For that purpose we use both perturbation theory results, extended here for a top-hat filter, and numerical simulations. In particular, we derive the quantitative (complete as possible up to third-order terms) and qualitative predictions for constrained averages and constrained dispersions – which describe the non-linearities and the stochasticity properties beyond the linear regime – and compare them against numerical simulations. We find overall a good agreement for constrained averages; however, the agreement for constrained dispersions is only qualitative. Scaling relations for the Ω-dependence of these quantities are satisfactorily reproduced. Guided by our analytical and numerical results, we finally construct a robust phenomenological description of the joint PDF in a closed analytic form. The good agreement of our formula with results of N-body simulations for a number of cosmological parameters provides a sound validation of the presented approach. Our results provide a basis for a potentially powerful tool with which it is possible to analyse galaxy survey data in order to test the gravitational instability paradigm beyond the linear regime and to put useful constraints on cosmological parameters. In particular, we show how the non-linearity in the density–velocity relation can be used to break the so-called Ω-bias degeneracy in cosmic density–velocity comparisons.

Coalescence and Star Formation in Collapsing Flattened Systems

We report on the results of N-body simulations of star formation resulting from mergers during the collapse of dynamically cold, flattened systems of cloudlets. Such evolution is expected to occur in several models of cluster star formation. As was found previously in the case of spherical systems, the resulting star clusters have half-mass radii that are significantly smaller than the initial values for the systems. Stars that form early in the collapse have large final orbital radii and retain a strong memory of the initial ellipticity of the system. Stars that form later have smaller final radii and retain less memory. If we examine only those stars within the final half-mass radii of the models, we find that they follow a much less ellipsoidal distribution than do stars at larger radii. Their ellipticities, however, generally exceed those of young, dynamically unevolved massive star clusters in the LMC. The best comparisons with the observed ellipticities are found in models for which the initial system minor-to-major axis ratios b/a≳0.2, system kinetic-to-potential energy ratios Q≳0.1, covering factors fτ≲0.2, or initial cloud masses Mi≪MG. In the above, fτ is the fraction of the projected area of the system covered by the cloudlets, and MG is the critical mass for gravitational instability of the cloudlets, which are assumed to evolve isothermally. The ratio Mi/MG is taken to be one of the model parameters. All of these ranges of parameters tend to delay the onset of star formation until late in the collapse, when the system is less elliptical. Systems with Mi≪MG tend to produce very flat stellar initial mass functions, much flatter than observed. Unless, therefore, fτ≲0.2 results from the fragmentation of the parent cloud, systems such as the young LMC clusters are unlikely to have formed from extremely flattened or dynamically cold initial conditions, even in the presence of dissipation during star formation.

Escape from a Crisis in Fokker-Planck Models

AbstractRecent N-body simulations have shown that there is a serious discrepancy between the results of N-body simulations and the results of Fokker-Planck simulations for the evolution of globular and rich open clusters under the influence of the galactic tidal field. In some cases, the lifetime obtained from Fokker-Planck calculations is more than an order of magnitude smaller than those from N-body simulations. In this paper we show that the principal cause for this discrepancy is an oversimplified treatment of the tidal field used in previous Fokker-Planck simulations. We performed new Fokker-Planck calculations using a more appropriate implementation for the boundary condition of the tidal field. The implementation is only possible with anisotropic Fokker-Planck models, while all previous Fokker-Planck calculations rely on the assumption of isotropy. Our new Fokker-Planck results agree well with N-body results. Comparison of the two types of simulations gives a better understanding of the evolution of such clusters.