Yet another fast multipole method without multipoles --- Pseudo-particle multipole method

In this paper we describe a new approach to implement the O(N) fast multipole method and $O(N\log N)$ tree method, which uses pseudoparticles to express the potential field. The new method is similar to Anderson's method, which uses the values of potential at discrete points to represent the potential field. However, for the same expansion order the new method is more accurate and computationally efficient.Comment: 14 pages, 2 figure

On the Origin of Density Cusps in Elliptical Galaxies

We investigated the dynamical reaction of the central region of galaxies to a falling massive black hole by N-body simulations. As the initial galaxy model, we used an isothermal King model and placed a massive black hole at around the half-mass radius of the galaxy. We found that the central core of the galaxy is destroyed by the heating due to the black hole and that a very weak density cusp ($\rho \propto r^{-\alpha}$, with $\alpha \sim 0.5$) is formed around the black hole. This result is consistent with recent observations of large elliptical galaxies with Hubble Space Telescope. The velocity of the stars becomes tangentially anisotropic in the inner region, while in the outer region the stars have radially anisotropic velocity dispersion. The radius of the weak cusp region is larger for larger black hole mass. Our result naturally explains the formation of the weak cusp found in the previous simulations of galaxy merging, and implies that the weak cusp observed in large elliptical galaxies may be formed by the heating process by sinking black holes during merging events.Comment: 14 pages with 29 EPS figures; LaTeX; new results added; accepted for publication in Ap

Post-Collapse evolution of globular clusters

A number of globular clusters appear to have undergone core collapse, in the sense that their predicted collapse time is much shorter than their current age. Simulations using gas models and Fokker-Planck approximation have shown that the central density of a globular cluster after the collapse undergoes nonlinear oscillation with large amplitude (gravothermal oscillation). However, whether such an oscillation actually takes place in a real $N$-body system has remained unsolved, because an $N$-body simulation with a sufficiently high resolution would have required the computing resource of the order of several Gflops$\cdot$years. In the present paper, we report the result of such a simulation, performed on a dedicated special-purpose computer GRAPE-4. We 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 confirmed that gravothermal oscillation takes place in an $N$-body system. The expansion phase shows all signatures that are considered as the evidences of the gravothermal nature of the oscillation. At the maximum expansion, the core radius is $\sim 1$\% of the half-mass radius for the run with 32,768 particles. The maximum core size $r_c$ depends on $N$, as $\propto N^{-1/3}$.Comment: To appear in Apj, Vol 470, Oct 20, 11 pages, 7 figures, Postscript (or GIF) version available at http://grape.c.u-tokyo.ac.jp/pub/people/makino/papers/osclong.ps or http://grape.c.u-tokyo.ac.jp/pub/people/makino/papers/osclong.htm

Direct Simulation of Dense Stellar Systems with GRAPE-6

In this paper we describe the current status of the GRAPE-6 project to develop a special-purpose computer with a peak speed exceeding 100 Tflops for the simulation of astrophysical N-body problems. One of the main targets of the GRAPE-6 project is the simulation of dense stellar systems. In this paper, therefore, we overview the basic algorithms we use for the simulation of dense stellar systems and their characteristics. We then describe how we designed GRAPE hardwares to meet the requirements of these algorithms. GRAPE-6 will be completed by the year 2001. As an example of what science can be done on GRAPE-6, we describe our work on the galactic center with massive black holes performed on GRAPE-4, the predecessor of GRAPE-6.Comment: 12 pages, to appear in Dynamics of Star Clusters and the Milky Way, eds. R. Spurzem et al. (ASP Conference Series