GRAPE is a project for a special-purpose computer dedicated to classical many-body problems. Its primary use is for gravitational many-body problem of stellar systems. It was initiated in 1989, and now its performance is in the range of 100 Tera-flops equivalent. As an introduction to this symposium, development of the GRAPE project is reviewed, where motivation to it is the main concern of this paper rather than the computer itself.

I'll briefly overview the present status and the future of the GRAPE project. GRAPE (GRAvity PiPE) project is a project to design, develop and use special-purpose computers for astrophysical N-body simulations to do large-scale N-body simulations. Our first machine, GRAPE-1 was completed in 1989 and offered the speed of 240 Mflops. Since then, we have continued to develop newer and faster machines, and the newest machine, the GRAPE-6, has achieved the peak speed of 32 Tflops. I'll briefly discuss GRAPE-6 and its parallel architecture, and then discuss the possible form of GRAPE-7, the next generation machines.

Accretion from many small planetesimals to planets is reviewed. Solid protoplanets accrete through runaway and oligarchic growth until they become isolated. The isolation mass of protoplanets in terrestrial planet region is about 0.1-0.2 Earth mass, which suggests giant impacts among the protoplanets in the final stage of terrestrial planet formation. On the other hand, the isolation mass in Jupiter's and Saturn's orbits is about a few to 5 Earth masses, which may be massive enough to trigger gas accretion onto the cores. The isolation mass in Uranus and Neptune's orbits is as large as their present cores. Extending the above arguments to extrasolar planetary systems that are formed from disks with various initial masses, we also discuss diversity of extrasolar planetary systems.

We investigate the evolution of a circumterrestrial disk of debris generated by a giant impact on the Earth and the dynamical characteristics of the moon accreted from the disk by using N-body simulation. We find that in most cases the disk evolution results in the formation of a single large moon on a nearly circular orbit close to the equatorial plane of the initial disk just outside the Roche limit. The efficiency of incorporation of disk material into a moon is 10-55%, which increases with the initial specific angular momentum of the disk. These results hardly depend on the initial condition of the disk as long as the disk mass is a few times the present lunar mass and most disk mass exists inside the Roche limit.

A self-consistent system of the Boltzmann equation and the Poisson equation is used to study the dynamical evolution of Saturn's main A, B, and C rings. The theory, as applied to the Saturnian ring system, predicts for several features, such as numerous irregular density wakes, with size and spacing between them of the order 4πρ ≈ 2πh, where ρ is the mean epicycle radius of the particle and h is the typical thickness of the system under study. In Saturn's rings, ρ ≲ 10 m. Computer N-body experiments are desribed which test the validities of the theory. Use of the 112-processor SGI Origin 2000 supercomputer is enabled us to make long runs using a large number of particles in the direct simulation code and thus simulate phenomena not previously studied numerically. We predict that forthcoming in 2004 Cassini spacecraft high-resolution images will reveal this recurrent fine-scale ∼ 100 m or so structure in low and moderately high optical depth regions of the rings.

Star clusters are remarkably efficient (relative to the field) at making type Ia supernovae candidates: very short period, massive double-white-dwarf stars and giant-white dwarf binaries. The high frequency of these systems is the result of dynamical encounters between (mostly) primordial binaries and other cluster stars. Orbital hardening rapidly drives the degenerate binaries to periods under ∼ 10 hours. Gravitational radiation emission and mergers then produce supra-Chandrasekhar objects in less than a Hubble time.

We also find that free-floating planets can remain bound to a star cluster for much longer than was previously assumed: of the order of the cluster half-mass relaxation timescale as opposed to the crossing-time. The planets in our N-body study are of Jupiter mass and are initially placed in circular orbits of between 0.05 and 50 AU about a parent star whose mass is chosen from a realistic initial mass function. This result is important in the context of the preliminary detection of a population of free-floating sub-stellar objects in the globular cluster M22.

Star formation is intimately linked to the dynamical evolution of molecular clouds. Turbulent fragmentation determines where and when protostellar cores form, and how they contract and grow in mass via accretion from the surrounding cloud material. Using numerical models of self-gravitating supersonic turbulence, efficiency, spatial distribution and timescale of star formation in turbulent interstellar clouds are estimated. Turbulence that is not continuously replenished or that is driven on large scales leads to a rapid formation of stars in a clustered mode, whereas interstellar turbulence that carries most energy on small scales results in isolated star formation with low efficiency. The clump mass spectrum for models of pure hydrodynamic turbulence is steeper than the observed one, but gets close to it when gravity is included. The mass spectrum of dense cores is log-normal for decaying and large-wavelength turbulence, similar to the IMF, but is too flat in the case of small-scale turbulence. The three-dimensional models of molecular cloud fragmentation can be combined with dynamical pre-main sequence stellar evolution calculations to obtain a consistent description of all phases of the star formation process. First results are reported for a one solar mass protostar.

I contend that impulsively triggered star formation is at least as important as spontaneous quasistatic star formation regulated by ambipolar diffusion, and probably more so. To support this contention, I describe and discuss SPH simulations of (i) star formation triggered by colliding clouds and expanding nebulae (HII regions, stellar-wind bubbles, and supernova remnants); (ii) violent interactions between protostellar discs and the resulting genesis of binary systems and low-mass companions; and (iii) protostellar collapse induced by a sudden increase in external pressure.

We review what has been learned recently using N-body simulations about the evolution of globular clusters. While simulations of star clusters have become more realistic, and now include the evolution of single and binary stars, the prospect of reaching large enough N is still a distant one. Nevertheless more restricted kinds of simulations have recently brought valuable progress for certain problems of current observational interest, including the origin and structure of tidal tails of globular clusters. In addition, such simulations have forced us to rethink some basic aspects of stellar dynamics, including, in particular, the process of escape. Finally we turn to faster, approximate methods for studying star cluster dynamics, where the role of N-body simulations is one of calibration.

The impact of rotation on the relaxational dynamics of dense stellar systems is reviewed; while in the past linear stability analysis and a few N-body models existed only, we report recent advances in the technique of 2D Fokker-Planck models for axisymmetric rotating star cluster, now extending into the post-core collapse phase. It is confirmed that rotating clusters, whether they are in a tidal field or not, evolve significantly faster in pre-collapse than non-rotating ones, while in post-collapse only those in a tidal field keep their larger speed of evolution. Consequences for observed shapes, density distribution, and kinematic properties of young and old star clusters are discussed.

We summarise the specification and some previously unpublished results of the collaborative experiment (“Kyoto I”) that was carried out at the time of the Kyoto General Assembly of the IAU. As the subject has advanced considerably since then, and codes have become more elaborate, we describe an agreed proposal for a new experiment ("Kyoto II").

The Colour-Magnitude diagram (CMD) is an important observational tool for use in the study of star clusters. Thus it is important to include a consistent treatment of stellar and binary evolution in N-body codes to allow the generation and interaction of the full range of stellar populations. In this way realistic cluster models can be compared with observed cluster populations via the respective CMDs. Preliminary results of such an approach using the Aarseth NBODY4 code on the GRAPE-6 are presented here. In particular, by using a stellar evolution algorithm that includes metallicity as a free parameter, the stellar populations of clusters at various stages of evolution can be studied. Interesting formation cases directly influenced by the cluster environment are highlighted for various stellar sub-populations.

A new formalism for studying three-body interactions is discussed. It introduces the concept of forced modes of oscillation of a binary, and relies on the chaos theory concept of resonance overlap. The treatment provides a powerful tool for studying stability and scattering in hierarchical systems, and is not restricted by mass ratios, eccentricities or orientations.

The past few years have seen dramatic improvements in the scope and realism of star cluster simulations. Accurate treatments of stellar evolution, coupled with robust descriptions of all phases of binary evolution, have been incorporated self-consistently into several dynamical codes, allowing for the first time detailed study of the interplay between stellar dynamics and stellar physics. The coupling between evolution, dynamics, and the observational appearance of the cluster is particularly strong in young systems and those containing large numbers of primordial binary systems, and important inroads have been made in these areas, particularly in N-body simulations. I discuss some technical aspects of the current generation of N-body integrators, and describe some recent results obtained using these codes.

We study the growth rate of stars in dense star clusters by stellar collisions. Our analytic calculations are in good agreement with direct N-body simulations with up to 65536 stars performed on the GRAPE family of computers. We find that star clusters with a half mass relaxation time ≲ 20 Myr are dominated by stellar collisions. The first collision occurs at the moment of core collapse. The collision dominated phase last until the cluster dissolves in the tidal field of the Galaxy or mass loss by stellar evolution arrests core collapse. The majority of collisions occur with the same star resulting in the uncontrolled growth of a super massive object. This object can grow up to ∼ 0.08% of the mass of the entire star cluster. This mass ratio is comparable to the ratio of the mass of Galactic bulges to their central black hole. Star clusters which are older than about 4 Myr and with a half mass relaxation time ≲ 80 Myr are expected to contain the remnant of a phase of uncontrolled growth in their cores.

Observations of stellar kinematics, gas dynamics and masers around galactic nuclei have now firmly established that many galaxies host central supermassive black holes (SMBHs) with masses in the range 106 ∼ 109M⊙. However, how these SMBHs formed is not well understood. One reason for this situation is the lack of observations of intermediate-mass BHs (IMBHs), which could bridge the gap between stellar-mass BHs and SMBHs. Recently, this missing link (i.e., an IMBH) has been found in observations made by ASCA and Chandra of the central region of the starburst galaxy M82 (Matsumoto and Tsuru 1999, Ptak and Griffith 1999, Matsumoto et al. 2001, Kaaret et al. 2001). Subsequent observations by SUBARU have revealed that this IMBH apparently coincides with a young compact star cluster. Based on these findings, we propose a new formation scenario for SMBHs. In this scenario, IMBHs first form in young compact star clusters through runaway merging of massive stars. While these IMBHs are forming, the host star clusters sink toward the galactic nucleus through dynamical friction, and upon evaporation deposit their IMBHs near the galactic center. The IMBHs then form binaries and eventually merge via gravitational radiation, forming an SMBH.