Evolution of Clusters of Galaxies: Mass Stripping from Galaxies and Growth of Common Halos

We investigated the evolution of clusters of galaxies using self-consistent $N$-body simulations in which each galaxy was modeled by many particles. We carried out simulations for about 20 cases using different initial conditions. In all simulations, clusters were initially in virial equilibrium. We found that more than half of the total mass escaped from individual galaxies within a few crossing times of the cluster, and that a diffuse halo was formed. The growth rate of the common halo depended on the size of individual galaxies only weakly. The stripping of the mass from galaxies was mainly due to the interaction of galaxies, not due to the effect of the tidal field of the cluster potential. The amount of stripped mass was larger for galaxies in the central region than for those in the outer region, since the interactions were more frequent in the central region. As a result, a positive correlation between the distance from the center and the mass of the galaxy developed. The volume-density profile of the common halo is expressed as $\rho\propto r^{-1}$ in the central region. This mass distribution is consistent with the mass distribution in clusters estimated using X-ray observations.Comment: 12 pages with 12 figures; accepted for publication in PAS

BRIDGE: A Direct-tree Hybrid N-body Algorithm for Fully Self-consistent Simulations of Star Clusters and their Parent Galaxies

We developed a new direct-tree hybrid N-body algorithm for fully self-consistent N-body simulations of star clusters in their parent galaxies. In such simulations, star clusters need high accuracy, while galaxies need a fast scheme because of the large number of the particles required to model it. In our new algorithm, the internal motion of the star cluster is calculated accurately using the direct Hermite scheme with individual timesteps and all other motions are calculated using the tree code with second-order leapfrog integrator. The direct and tree schemes are combined using an extension of the mixed variable symplectic (MVS) scheme. Thus, the Hamiltonian corresponding to everything other than the internal motion of the star cluster is integrated with the leapfrog, which is symplectic. Using this algorithm, we performed fully self-consistent N-body simulations of star clusters in their parent galaxy. The internal and orbital evolutions of the star cluster agreed well with those obtained using the direct scheme. We also performed fully self-consistent N-body simulation for large-N models ($N=2\times 10^6$). In this case, the calculation speed was seven times faster than what would be if the direct scheme was used.Comment: 12 pages, 13 figures, Accepted for PAS

Time-Symmetrized Kustaanheimo-Stiefel Regularization

In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although analytical solutions exist in the unperturbed and weakly perturbed cases, numerical integration is necessary in situations where the perturbation is relatively strong. Kustaanheimo--Stiefel (KS) regularization is widely used to remove the singularity in the equations of motion, making it possible to integrate orbits having very high eccentricity. However, even with KS regularization, long-term integration is difficult, simply because the required accuracy is usually very high. We present a new time-integration algorithm which has no secular error in either the binding energy or the eccentricity, while allowing variable stepsize. The basic approach is to take a time-symmetric algorithm, then apply an implicit criterion for the stepsize to ensure strict time reversibility. We describe the algorithm in detail and present the results of numerical tests involving long-term integration of binaries and hierarchical triples. In all cases studied, we found no systematic error in either the energy or the angular momentum. We also found that its calculation cost does not become higher than those of existing algorithms. By contrast, the stabilization technique, which has been widely used in the field of collisional stellar dynamics, conserves energy very well but does not conserve angular momentum.Comment: figures are available at http://grape.c.u-tokyo.ac.jp/~funato/; To appear in Astronomical Journal (July, 1996

Light to Mass Variations with Environment

Large and well defined variations exist between the distribution of mass and the light of stars on extragalactic scales. Mass concentrations in the range 10^12 - 10^13 M_sun manifest the most light per unit mass. Group halos in this range are typically the hosts of spiral and irregular galaxies with ongoing star formation. On average M/L_B ~ 90 M_sun/L_sun in these groups . More massive halos have less light per unit mass. Within a given mass range, halos that are dynamically old as measured by crossing times and galaxy morphologies have distinctly less light per unit mass. At the other end of the mass spectrum, below 10^12 M_sun, there is a cutoff in the manifestation of light. Group halos in the range 10^11 - 10^12 M_sun can host dwarf galaxies but with such low luminosities that M/L_B values can range from several hundred to several thousand. It is suspected that there must be completely dark halos at lower masses. Given the form of the halo mass function, it is the low relative luminosities of the high mass halos that has the greatest cosmological implications. Of order half the clustered mass may reside in halos with greater than 10^14 M_sun. By contrast, only 5-10% of clustered mass would lie in entities with less than 10^12 M_sun.Comment: 15 pages, 9 figures, 2 tables, Accepted Astrophysical Journal 619, 000, 2005 (Jan 1

Collisional Evolution of Galaxy Clusters and the Growth of Common Halos

We investigated the dynamical evolution of clusters of galaxies in virial equilibrium using Fokker-Planck models and self-consistent N-body models. In particular, we focused on the growth of a common halo, which is a cluster-wide halo formed by matter stripped from galaxies, and the development of a central density cusp. The Fokker-Planck models include the effects of two-body gravitational encounters both between galaxies and between galaxies and common halo particles. The effects of tidal mass stripping from the galaxies due to close galaxy-galaxy encounters and accompanying dissipation of the orbital kinetic energies of the galaxies were also taken into account in the Fokker-Planck models. We find that the results of the Fokker-Planck models are in excellent agreement with those of the N-body models regarding the growth of the common halo mass and the evolution of the cluster density profiles. In the central region of the cluster, a shallow density cusp, approximated by $\rho (r) \propto r^{-\alpha}$ ($\alpha \sim$ 1), develops. This shallow cusp results from the combined effects of two-body relaxation and tidal stripping. The cusp steepness, $\alpha$, weakly depends on the relative importance of the tidal stripping. When the effect of stripping is important, the central velocity dispersion decreases as the central density increases and, consequently, a shallow ($\alpha <2$) cusp is formed. In the limit of no stripping, usual gravothermal core collapse occurs, i.e. the central velocity dispersion increases as the central density increases with a steep ($\alpha >2$) cusp left. We conclude from our consideration of the origin of the cusp demonstrated here that shallow cusps should develop in real galaxy clusters.Comment: revised, 21 pages, 16 figures, to appear in PASJ, 54, No.1 (2002

Evolution of Star Clusters near the Galactic Center: Fully Self-consistent N-body Simulations

We have performed fully self-consistent $N$-body simulations of star clusters near the Galactic center (GC). Such simulations have not been performed because it is difficult to perform fast and accurate simulations of such systems using conventional methods. We used the Bridge code, which integrates the parent galaxy using the tree algorithm and the star cluster using the fourth-order Hermite scheme with individual timestep. The interaction between the parent galaxy and the star cluster is calculate with the tree algorithm. Therefore, the Bridge code can handle both the orbital and internal evolutions of star clusters correctly at the same time. We investigated the evolution of star clusters using the Bridge code and compared the results with previous studies. We found that 1) the inspiral timescale of the star clusters is shorter than that obtained with "traditional" simulations, in which the orbital evolution of star clusters is calculated analytically using the dynamical friction formula and 2) the core collapse of the star cluster increases the core density and help the cluster survive. The initial conditions of star clusters is not so severe as previously suggested.Comment: 19 pages, 19 figures, accepted for publication in Ap