Piet Hut: IAU Symp. 174 Panel Discussion

Panel Discussion of IAU Symp. 174 on `Dynamical Evolution of Star Clusters'

Contribution by Piet Hut

Theoretical studies of dense stellar systems have recently seen rapid progress in the development and refinement of many different techniques for modeling star systems. I will briefly mention six classes of techniques, while commenting on the progress to be expected therein during the years to come, until the next IAU Symposium on star cluster dynamics.

1. N-Body Techniques. Until the time of this meeting, N-body techniques did not go much further than producing toy models. These were certainly interesting and promising, but fell far short of modeling realistic globular clusters. The sheer computational expense was simply too forbidding to come even close to modeling clusters on a star-by-star basis, given that most globular clusters have numbers of stars in the range 100,000 - 1,000,000. All this has changed, now that the GRAPE-4 special purpose computer has come online, with its Teraflops speed (see Taiji's contribution). As detailed in Makino's paper, calculations with more than 30,000 particles have now become routine, and calculations with up to 100,000 particles are feasible with the new generation of front end computers that will become available in 1996. In addition, the GRAPE designers are now setting their eyes on an even more ambitious goal, namely the development of a machine with a speed a factor 1000 higher than that of the current GRAPE-4. Such a Petaflops computer could become operational by the year 2000, if the necessary funding will be found, something that is being actively pursued at present.

2. Scattering Experiments. On the other side of the spectrum of possibilities for star-by-star modeling of processes in globular clusters, we find local treatments of the `microphysics' of close encounters between single stars and binaries. Significant progress has been made in this area, for example through the development of fully automated scattering software. Specification of a few physical parameters is sufficient to start up this software laboratory, allowing the set-up, execution, and on-line analysis of experiments to be carried out without any human intervention. Extensions of this package to include binary-binary scattering is currently underway; see McMillan's contribution to these proceedings for further details. Other approaches to the dynamics of small-N systems include a stability analysis of hierarchical triples (see Kiselova's contribution), the addition of hydrodynamical effects (Davies' paper), and an analysis of the frequency of physical collisions during scattering events (Chernoff and Huang's contribution).

3. Numerical Approximation Techniques. Until this year, simulations of globular clusters with realistic particle numbers of 100,000 and larger could only be undertaken through a variety of numerical approximation techniques. Between twenty and thirty years ago, various Monte Carlo approaches have been developed for solving the evolution of a star cluster in the Fokker-Planck approximation. Soon afterwards, direct integration techniques have been developed for solving the Fokker-Planck equation. In addition, various conducting gas sphere models have been developed. The combination of all these models, and detailed comparative studies of the results of the different techniques, have proved beneficial for our understanding of the strengths and weaknesses, as well as the limits of applicability, of the individual models. For detailed descriptions, see the contributions in these proceedings by Giersz, Lee, McMillan and Engle, Spurzem, and Takahashi, as well as the talk by Heggie, who reports on an interesting technique in which the results from many small-N runs are averaged. Significant further progress can be expected, both in the ongoing development and refinement of individual techniques, as well as in applications in connection with the new GRAPE hardware. Even though realistic N-body calculations now are feasible, allowing a star-by-star modeling of globular clusters, these calculations still remain expensive, requiring sometimes weeks and often months for a single run to reach completion. Approximate methods therefore are called upon in order to interpolate between, and perhaps even extrapolate from, the few detailed runs coming out of the GRAPE machinery.

4. Analytic Techniques. Although one sometimes gets the impression that current research in star cluster dynamics is almost completely a numerical enterprize, there is plenty of room left for the development of new analytic techniques. Perhaps the most promising area will be that of the development of physically inspired fitting formulas. Rather than tabulating the results of detailed numerical calculations, or applying arbitrary curve fitting to those results, it is often possible to use more physically motivated reasoning in the choice of fitting functions. This has the double advantage of allowing more physical insight in the results obtained as well as providing a measure of confidence in extrapolations beyond the regimes currently tested numerically. An example of this approach is presented in the poster paper by Heggie {\it et al.}. Others examples of analytical approximation techniques are given in the contribution by Mardling and in the posters by Heggie, and Heggie and Rasio.

5. Approximate Treatments of Stellar Evolution. In addition to the four techniques listed above, which can be applied to the purely gravitational N-body problem, a more realistic treatment of star cluster evolution requires us to go beyond the point-mass approximation. A first step in this direction is the inclusion of a time dependency of the mass of a single star, for example by taking into account the mass that is lost in later stages of stellar evolution, towards the formation of a white dwarf, neutron star, or black hole. However, as soon as this step has been taken for single stars, far more complicated issues arise when we want to apply such simple recipes to the case of interacting binary stars, let alone the simultaneous interaction of three or four stars during scattering encounters. How to treat mass overflow, how to discriminate between stable and unstable cases of such overflow, what to do with common envelope phases in their evolution --- all these questions require careful consideration, together with the imperative to refrain from too-fancy a type of solution. The challenge to produce a coherent set of recipes that are as simple as possible, but no simpler, across the board is a formidable one. This challenge has recently been taken up by several individuals, as can be seen from the contributions by Aarseth, Eggleton, and Portegies Zwart.

6. Approximate Treatments of Hydrodynamics. Another extension that is called for in realistic star cluster modeling, beyond the point-mass limit, is the inclusion of some type of hydrodynamics. Smooth Particle Hydrodynamics forms a natural candidate; see the papers by Davies and Rasio. In the not-too-distant future, it will be feasible to include local SPH calculations as an option in large N-body calculations, since the additional computational cost required will become relatively less for larger N values. The challenges in combining stellar dynamics and SPH techniques will mainly take the form of software technicalities, related to the complicated bookkeeping required for the treatment of simultaneous three-body and four-body interactions, and occasional interactions with much larger number of particles. When following 10^5 stars for a Hubble time, exceptional cases are bound to occur in which, for example, two binaries will be involved in a complex resonance encounter while at the same time encountering yet another binary. The treatment of the inclusion of stars into and escape from such a six-star interaction will be far from straightforward, and has not yet been attempted in all generality.

Other techniques could be mentioned here as well, especially when we include the modeling of dense galactic nuclei. Here the possible presence of black holes, as well as young stars and molecular clouds, create additional complications. Closer to home, the modeling of a proto-planetary nebula again sets different requirements for the physics to be included in a study of the dynamics of this type of dense stellar systems. However, the above six classes of techniques, while not being exhaustive, give at least some taste of the further developments to be expected in the near future.

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