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Million-Body Problem: Star Clusters

For decades, the holy grail of star cluster simulations has been to perform a direct N-body calculation for the ten-billion year history of a globular star cluster, modeling the stars on a one-to-one basis. This implies solving the 1,000,000-body problem with staggering discrepancies in length and time scales. Neutronstars and black holes can form binary systems or ondergo unbound encounters with relevant time scales of milliseconds and length scales of kilometers. Given that globular clusters have dimensions of hundreds of light years and ages of more than ten billion years, the range in length scales spans more than fifteen orders of magnitude while the range in time scales exceeds twenty orders of magnitude. All this puts severe constraints on both hardware and software used in such simulations.

We are now in a position to carry out these types of realistic simulations. Progress may have seemed slow: we saw simulations with tens of stars in the sixties, hundreds in the seventies, thousands in the eighties, and finally tens of thousands in the nineties. The reason for this modest advance is that the computational cost of a full star cluster evolution scales proportional to the third power of the number of particles: two powers for the inter-particle interactions at each time step, and an additional power because the increase in particle number slows down the heat flow time by making two-body relaxation less effective.

Following the previous trend, we expect to see routine simulations of star clusters using several hundred of thousands of particles in the twothousandzeroes. The main challenge will now be to include enough physics to make these simulations sufficiently realistic to compare them in detail with the increasingly comprehensive observations of star clusters.

Dynamical Evolution

The first challenge is to model and interpret the rich set of phenomena already present in the purely gravitational N-body problem. Surprisingly, each increase of an order of magnitude in the number of stars has shown a qualitative chance in the properties of star cluster simulations, even though the underlying Newtonian dynamics is completely scale-free. There are various phenomena responsible for this, such as the occurrence of instabilities of a gravothermal nature, and the fact that binary star properties scale differently from single star properties. Here is a list of some of my papers that address problems in pure gravity:

Primordial Binaries

In the late eighties, based on various types of observations, it became clear that for most globular star clusters a significant fraction of their stars were part of primordial binary systems. Until then, almost all simulations of star cluster evolution had started with a collection of single stars, so it was time to go back to the drawing boards. Our first sketch of the implications for globular cluster evolution appeared in the paper: We then carried out several 1000-body simulations containing significant fractions of primordial binaries: Quite a bit later, we extended these calculations to include much larger number of particles, and also a much larger number of runs: We published a brief review as: while we also used more approximate methods, in: Finally, we published a detailed survey review of observational and well as theoretical developments concerning primordial binaries in:

X-Ray Binaries

It was the discovery of an abundance of X-ray sources in globular clusters, in the early seventies, that changed our picture of those clusters. Rather than just being old and boring remnants of the early formation phases of our galactic environment, globular clusters were seen to be laboratories for rather unusual experiments in stellar evolution. It became clear that the high stellar densities in the centers of the clusters were responsible for the formation of X-ray sources, although many of the details long remained uncertain. Today, too, properties of globular cluster X-ray sources remain among the prime diagnostics for their evolutionary histories. Here are some papers we wrote on the subject:

Here is a summary of strong observational evidence that X-ray sources in globular clusters are indeed formed through dynamical encounters:

Following up on this paper, we showed that for cataclysmic variables, too, the majority has a dynamical origin: