arXiv subjects

Junichiro Makino

Publications and source records attributed to Junichiro Makino.

125 records · Page 7Linked to original sources

arXiv1999

GRAPE-5: A Special-Purpose Computer for N-body Simulation

We have developed a special-purpose computer for gravitational many-body simulations, GRAPE-5. GRAPE-5 is the successor of GRAPE-3. Both consist of eight custom pipeline chips (G5 chip and GRAPE chip). The difference between GRAPE-5 and GRAPE-3 are: (1) The G5 chip contains two pipelines operating at 80 MHz, while the GRAPE chip had one at 20 MHz. Thus, the calculation speed of the G5 chip and that of GRAPE-5 board are 8 times faster than that of GRAPE chip and GRAPE-3 board. (2) The GRAPE-5 board adopted PCI bus as the interface to the host computer instead of VME of GRAPE-3, resulting in the communication speed one order of magnitude faster. (3) In addition to the pure 1/r potential, the G5 chip can calculate forces with arbitrary cutoff functions, so that it can be applied to Ewald or P^3M methods. (4) The pairwise force calculated on GRAPE-5 is about 10 times more accurate than that on GRAPE-3. On one GRAPE-5 board, one timestep of 128k-body simulation with direct summation algorithm takes 14 seconds. With Barnes-Hut tree algorithm (theta = 0.75), one timestep of 10^6-body simulation can be done in 16 seconds.

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arXiv1999

PROGRAPE-1: A Programmable, Multi-Purpose Computer for Many-Body Simulations

We have developed PROGRAPE-1 (PROgrammable GRAPE-1), a programmable multi-purpose computer for many-body simulations. The main difference between PROGRAPE-1 and "traditional" GRAPE systems is that the former uses FPGA (Field Programmable Gate Array) chips as the processing elements, while the latter rely on the hardwired pipeline processor specialized to gravitational interactions. Since the logic implemented in FPGA chips can be reconfigured, we can use PROGRAPE-1 to calculate not only gravitational interactions but also other forms of interactions such as van der Waals force, hydrodynamical interactions in SPH calculation and so on. PROGRAPE-1 comprises two Altera EPF10K100 FPGA chips, each of which contains nominally 100,000 gates. To evaluate the programmability and performance of PROGRAPE-1, we implemented a pipeline for gravitational interaction similar to that of GRAPE-3. One pipeline fitted into a single FPGA chip, which operated at 16 MHz clock. Thus, for gravitational interaction, PROGRAPE-1 provided the speed of 0.96 Gflops-equivalent. PROGRAPE will prove to be useful for wide-range of particle-based simulations in which the calculation cost of interactions other than gravity is high, such as the evaluation of SPH interactions.

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arXiv1999

Thermal Relaxation in One-Dimensional Self-Gravitating Systems

In this paper, we study the thermal relaxation in the one-dimensional self-gravitating system, or the so-called sheet model. According to the standard argument, the thermal relaxation time of the system is around $Nt_c$, where $N$ is the number of sheets and $t_c$ is the crossing time. It has been claimed that the system does not reach the thermal equilibrium in this thermal relaxation timescale, and that it takes much longer time for the system to reach true thermal equilibrium. We demonstrate that this behavior is explained simply by the fact that the relaxation time is long. The relaxation time of sheets with average binding energy is $\sim 20 Nt_c$, and that of sheets with high energy can exceed $1000 Nt_c$. Thus, one needs to take the average over the relaxation timescale of high-energy sheets, if one wants to look at the thermal characteristic of these high energy sheets.

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arXiv1999

Star cluster ecology III: Runaway collisions in young compact star clusters

The evolution of young compact star clusters is studied using N-body simulations in which both stellar evolution and physical collisions between stars are taken into account. The initial conditions are chosen to represent R136, a compact star cluster in the 30 Doradus region of the Large Magellanic Cloud. The present runs do not include the effects of primordial binaries. We find that physical collisions between stars in these models are frequent, and that the evolution of the most massive stars and the dynamical evolution of the cluster are closely coupled. In all cases, a single star grows steadily in mass through mergers with other stars, forming a very massive (>100 Msun) star in less than 3-4 Myr. The growth rate of this runaway merger is much larger than estimates based on simple cross-section arguments, mainly because the star is typically found in the core and tends to form binaries with other massive stars there. The runaway is ``rejuvenated'' by each new collision, and its lifetime is extended considerably as a consequence. Observationally, such a star will appear in the Hertzsprung-Russell diagram as a blue straggler. When the runaway forms a black hole, the binary in which it is found is usually dissociated.

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arXiv1998

A Simple Formulation of the Fast Multipole Method: Pseudo-Particle Multipole Method

We present the pseudo-particle multipole method (P2M2), a new method to handle multipole expansion in fast multipole method and treecode. This method uses a small number of pseudo-particles to express multipole expansion. With this method, the implementation of FMM and treecode with high-order multipole terms is greatly simplified. We applied P2M2 to treecode and combined it with special-purpose computer GRAPE. Extensive tests on the accuracy and calculation cost demonstrate that the new method is quite attractive.

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arXiv1998

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 ($ρ\propto r^{-α}$, with $α\sim 0.5$) 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.

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arXiv1998

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.

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arXiv1998

On the dissolution of evolving star clusters

Using direct N-body simulations which include both the evolution of single stars and the tidal field of the parent galaxy, we study the dynamical evolution of globular clusters and rich open clusters. We compare our results with other N-body simulations and Fokker-Planck calculations. Our simulations, performed on the GRAPE-4, employ up to 32,768 stars. The results are not in agreement with Fokker-Planck models, in the sense that the lifetimes of stellar systems derived using the latter are an order of magnitude smaller than those obtained in our simulations. For our standard run, Fokker-Plank calculations obtained a lifetime of 0.28 Gyr, while our equivalent N-body calculations find about 4 Gyr. The principal reason for the discrepancy is that a basic assumption of the Fokker-Plank approach is not valid for typical cluster parameters. The stellar evolution timescale is comparable to the dynamical timescale, and therefore the assumption of dynamical equilibrium leads to an overestimate of the dynamical effects of mass loss. Our results suggest that the region in parameter space for which Fokker-Planck studies of globular cluster evolution, including the effects of both stellar evolution and the galactic tidal field, are valid is limited. The discrepancy is largest for clusters with short lifetimes.

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arXiv1997

Time Symmetrization Meta-Algorithms

We present two types of meta-algorithm that can greatly improve the accuracy of existing algorithms for integrating the equations of motion of dynamical systems. The first meta-algorithm takes an integrator that is time-symmetric only for constant time steps, and ensures time-symmetry even in the case of varying time steps. The second meta-algorithm can be applied to any self-starting integration scheme to create time-symmetry, both for constant and for variable time steps, even if the original scheme was not time-symmetric. Our meta-algorithms are most effective for Hamilton systems or systems with periodic solutions. If the system is not Hamiltonian (for example, if some dissipative force exists), our methods are still useful so long as the dissipation is small.

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arXiv1997

GRAPE-6: A Petaflops Prototype

We present the outline of a research project aimed at designing and constructing a hybrid computing system that can be easily scaled up to petaflops speeds. As a first step, we envision building a prototype which will consist of three main components: a general-purpose, programmable front end, a special-purpose, fully hardwired computing engine, and a multi-purpose, reconfigurable system. The driving application will be a suite of particle-based large-scale simulations in various areas of physics. The prototype system will achieve performance in the $\sim 50 - 100$ teraflops range for a broad class of applications in this area. The combination of a hardwired petaflops-class computational engine and a front end with sustained speed on the order of 10 gigaflops can produce extremely high performance, but only for the limited class of problems in which there exists a single bottleneck with computing cost dominating the total. While the calculation for which the Grape-4 (our system's immediate predecessor) was designed is a prime example of such a problem, in many other applications the primary computational bottleneck, while still related to an inverse-square (gravitational, Coulomb, etc.) force, requires less than 99% of the computing power. Although the remainder of the CPU time is typically dominated by just one secondary bottleneck, its nature varies greatly from problem to problem. It is not cost-effective to attempt to design custom chips for each new problem that arises. FPGA-based systems can restore the balance, guaranteeing scalability from the teraflops to the petaflops domain, while still retaining significant flexibility. (abbreviated abstract)

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arXiv1997

Merger Rate of Equal-Mass Spherical Galaxies

We present cross sections and reaction rates for merging to occur during encounters of equal-mass spherical galaxies. As an application, we determine the rate of galaxy merging in clusters of galaxies. We present results for two types of Plummer models (a full and a truncated one), two King models and the Hernquist model. Cross sections are determined on the basis of a large number ($\sim 500$) of simulations of galaxy encounters, using the 10-Gigaflops GRAPE 3A special-purpose computer. We characterize the overall merger rate of galaxies in a galaxy cluster by a single number, derived from our cross sections by an integration over galaxy encounter velocities in the limit of a constant density in velocity space. For small clusters, where the cluster velocity dispersion may not significantly exceed the internal velocity dispersion of the individual galaxies, this constant-density approximation may not be valid. For those cases, we present separate results, based on integrations of our cross sections over Maxwellian velocity distributions. Finally, tidal effects from the cluster potential as well as from neighboring galaxies may prevent a barely bound galaxy pair from spiraling in after their first encounter. We give a quantitative estimate of the resulting reduction in the actual merger rate, due to these tidal interactions.

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arXiv1996

On the Origin of Cusps in Dark Matter Halos

Observed cusps with density profiles $ρ\propto r^{-1}$ or shallower, in the central regions of galaxies, cannot be reproduced in the standard Cold Dark Matter (CDM) picture of hierarchical clustering. Previous claims to the contrary were based on simulations with relatively few particles, and substantial softening. We present simulations with particle numbers an order of magnitude higher, and essentially no softening, and show that typical central density profiles are clearly steeper than $ρ\propto r^{-1}$. The observed shallower profiles may have formed through the smoothing effect of the spiral-in of central black holes in previous merger phases. In addition, we confirm the presence of a temperature inversion in the inner 5 kpc of massive galactic halos, and illustrate its formation as a natural result of the merging of unequal progenitors.

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arXiv1996

N-body Simulation of Galaxy Formation on GRAPE-4 Special-Purpose Computer

We report on resent N-body simulations of galaxy formation performed on the GRAPE-4 (GRAvity PipE) system, a special-purpose computer for astrophysical N-body simulations. We review the astrophysical motivation, the algorithm, the actual performance, and the price per performance. The performance obtained is 332 Gflops averaged over 185 hours for a simulation of a galaxy formation with 786,400 particles. The price per performance obtained is 4,600 dollars per Gflops. The configuration used for the simulation consists of 1,269 pipeline processors and has a peak speed of 663 Gflops.

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arXiv1996

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}$.

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arXiv1996

Merging of galaxies with central black holes II. Evolution of the black hole binary and the structure of the core

We investigated the evolution of the black hole binary formed by the merging of two galaxies each containing a central massive black hole. Our main goal here is to determine if the black hole binary can merge through the hardening by dynamical friction and the gravitational wave radiation. We performed $N$-body simulations of merging of two galaxies with wide range of total number of particles to investigate the effect of the number of particles on the evolution of the black hole binary. We found that the evolution timescale was independent of the number of particles in the galaxy $N$ until the separation reaches a critical value. After the separation became smaller than this critical value, the evolution timescale was longer for larger number of particles. Qualitatively, this behavior is understood naturally the result the ``loss-cone'' effect. However, the dependence of the timescale on $N$ is noticeably weaker than the theoretical prediction. In addition, the critical separation is smaller than the theoretical prediction. The timescale of evolution through gravitational radiation at this critical separation is longer than the Hubble time. We discuss the reason of these discrepancy and the implication of the present result on the structure of the ellipticals and QSO activities.

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arXiv1994

The Effect of Gravitational Scattering on the Anisotropy of the Cosmic Background Radiation

The homogeneity of the cosmic microwave background radiation (CBR) is one of the most severe constraint for theories of the structure formation in the universe. We investigated the effect of the gravitational scattering (lensing) of galaxies, clusters of galaxies, and superclusters on the anisotropy of the CBR by numerical simulations. Although this effect was thought to be unimportant, we found that the gravitational scatterings by superclusters can significantly reduce the anisotropy of the CBR. We took into account the exponential growth of the distance between two rays due to multiple scatterings. The bending angle of each ray grows through the random walk process. On the other hand, difference between two rays grows exponentially while it is small. This exponential growth is caused by coherent scatterings that two rays suffer, and was neglected in the previous studies. The gravitational scattering by superclusters reduces the observed temperature anisotropy of the CBR at present time approximately by 40--60 $\%$ from that at the recombination time for angular scale up to a few degrees if the supercluster were formed at $z=$2--4.

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arXiv1994

Exponential Growth of Distance between Nearby Rays due to Multiple Gravitational Scatterings

We give an estimate of the relative error in the angular measurement of observations for high redshift objects induced by gravitational scatterings (lensing). Gunn (1967) concluded that the gravitational scatterings by galaxies induce the relative error of a few percent in the observations for objects at $z=1$. This estimate has been considered as a fundamental limitation of accuracy of the angular measurements in the observational cosmology. In multiple graviational scatterings, bending angle of single ray grows through the random work process. Gunn (1967) assumed that the difference of nearby rays also grows through the random walk process. However, distance between nearby photons grows exponentially because the two rays suffer coherent scatterings by the same scattering object. This exponential growth continues as long as the scattering is coherent. In the case of scattering by individual galaxies, the exponential growth continues until the angular distance reaches an arcminute or so. The relative error of the angular measurements under an arcminute due to the exponential growth is $\sim 30\%$ at $z=1$ and exceeds $100\%$ at $z=3$, in the case that the density parameter of galaxies is 0.2. The effects of clusters of galaxies or superclusters are more difficult to estimate accurately, but might be significant. In the case of supercluster the angular measurements up to a few degrees could be affected.

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