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1. Introduction

1.1. A Chat between Two Astronomers

Alice: Hi Bob, what are you up to?

Bob: Well, to be honest, I'm not quite sure yet. I'm in between projects right now. It's nice to sit back for a bit, looking at the whole field of astronomy, before plunging into a new project. But realistically speaking, it is likely I will continue working on star clusters. How about you?

Alice: Actually, I'm in a similar situation of having finished old projects and not yet taken on a major new one. Since I moved here, a month ago, I've been dealing with all kind of chores that had been accumulating, but now I have a clean desk, and I'm ready to start up.

Bob: The previous time I felt like this was when I had handed in my thesis. I was surprised to find myself suddenly in a pleasant vacuum, after rushing so much to get everything finished in time. It was soon afterward that I started to get involved in various parallel projects, which never seemed to come to an end. That was eight years ago.

Alice: I heard you just got tenure. I guess that has something to do with your finishing up your two latest projects?

Bob: how did you guess! While I enjoyed my work, I did feel a little constrained. There were times that I would have loved to go in some other directions, but that probably would not have been wise to do at that point.

Alice: I think you made the right decision. When I got tenure, about ten years ago now, I was in a different situation. We were in the middle of a large cosmological project, simulating the formation of large scale structure in the universe, working together with people from various teams. It was an exciting time, in which there were a number of basic questions that we could address for the first time.

Bob: I guess cosmology has now become as detailed a modeling job as any other field in astronomy?

Alice: Yeah. Now that the standard model is pretty well understood, cosmology has become a rather ordinary field in astronomy. But I never specialized in cosmology. While most of my work has been in stellar dynamics, I seem to keep moving between different fields. For example, I just finished a paper in planetary dynamics on the dynamical formation of binaries in the Kuiper Belt.

Bob: I'm certainly familiar with your publications in cluster dynamics, especially the analytical treatments you have given. They were quite useful for my more applied simulations. And didn't I see a paper of yours on black hole dynamics recently?

Alice: Yes, that was on the question of whether two massive black holes spiral in within a Hubble time, after they have been brought into proximity by the merging of their parent galaxy. And indeed, I tend to work on more analytical questions, though I enjoy doing large simulations too. Comparing the two and figuring out from both sides why there are discrepancies is most fun: it keeps surprising me that you can use pen and paper to predict roughly what a computer will come up with after performing a trillions of floating point calculations.

Bob: Which of all those fields in stellar dynamics do you find most interesting, having worked on all length scales between asteroids and cosmology?

Alice: Right now I would say star clusters, especially those star clusters where there is an appreciable chance for stellar collisions in the densest regions. Dense stellar systems, in other words.

Bob: As you know, this is close to my interest and background. But let me play the devil's advocate for a moment. What is so interesting about dense stellar systems, in particular?

1.2. Dense Stellar Systems

Alice: A quarter century ago, when I was an undergraduate, astronomy was much less unified than it is today. People observing in different wavelength bands did not talk as much with each other as they do today, and theorists studying stars, star clusters, galaxies, and cosmology had even less overlap in their research. Someone simulating the birth or evolution of a star; someone simulating a star cluster; someone simulating a collision between two galaxies; or someone studying large scale structure of the universe -- all four of them would be working separately, each on their own island. In contrast, we now see bridges everywhere.

When studying large scale structure simulations, we see how normal galaxies are built up through the process of merging smaller galaxies and protogalactic gas clouds. And when we study a collision between two galaxies, we see how star clusters are formed through the shocks that occur in the bridges and tails connecting them. So this already connects the largest three categories from the largest scale down. Similarly, when we start at the bottom, studying the formation of an individual star, we have learned that we can only understand the formation process in an environmental way, taking into account the interactions between many protostellar clouds simultaneously. So this forces us to study a whole star forming region, a proto stellar cluster. In this way all four fields are now connected.

It is fair to say that dense stellar systems form a central topic in the current trend toward unification of astrophysics. Not only is the topic connected with structure from the scale of individual stars up to that of galaxies and beyond, it also is connected with the study of extreme forms of matter, including neutron stars and black holes. And finally, on the level of simulations, it combines the simplest and most elegant theory of complexity, that of self-gravitational systems, with a diverse mix of astrophysical effects, as seen in stellar evolution and stellar hydrodynamics, including the physics of accretion disks, common envelope evolution, and so on.

Bob: That is all nice and fine, from a high-level point of view. However, when I'm working in the trenches, so to speak, I do not have much use for the big picture, and my only job is getting the details straight, and to get the work done.

Alice: Of course, the details are what counts, in any real piece of research, if you want to get any real work done. But it is equally important to keep sight of a broader picture. If not, you'll still get a lot of work done, but not necessarily in an efficient way, and it will not necessarily lead to interesting results.

Bob: I agree that you have to be sensible. But you can be sensible without grand pictures and declarations, I would think. How can a grand picture help me to get my work done?

Alice: There are many examples. Just to name one in stellar dynamics: the introduction of tree codes in the mid eighties made it possible to do the type of cosmological simulations that we are all familiar with. Until then, there was a huge gap between codes and direct summation codes. The breakthrough came not by improving details in a particular piece of work in a given project, but by stepping back and rethinking the whole approach to large-scale simulations in stellar dynamics.

Bob: I guess you could call that a paradigm shift. But such shifts are few and far between. I don't think you can take that as a typical example. What I see as the future of dense stellar systems simulations doesn't require any paradigm shift. Computers are getting fast enough, and what is needed is to write more software, of the type that can handle not only stellar dynamics but also stellar evolution, on the fly. And we can throw in hydrodynamics as well. In short: what I see happening is a type of `kitchen sink' approach to simulating dense stellar systems.

Alice: That doesn't sound very attractive. At the very least I hope you come up with a better name! But more seriously, why would you want to do that? And even if you were to do that, why would anyone believe the results from such a monstrous combination of complicated codes?

Bob: I think we have no other choice. I grant you that idealized abstractions have their use, by now we have learned enough about a system of self-gravitating point masses, all of equal mass, and it is time to move on.

Alice: Well, moving on beyond equal-mass point masses, and doing your kind of kitchen sink simulation, adding stellar evolution and hydrodynamics, those are quite different concepts. Surely there is a middle ground between the two!

Bob: Not really, I would say. It may sound strange, but I think there are only two reasonable ways to simulate a star cluster: either you use the most extreme idealization, in which all stars have the same mass, and with radius zero; or you couple every star to its own stellar evolution program, to model its internal degrees of freedom, at least in principle.

Alice: Strange indeed. Please explain.

1.3. The Equal-Mass Point-Particle Approximation

Bob: Let us first look at the history. I presume you know how star cluster modeling got started. You have seen more of it than I have.

Alice: When it all got started, I had barely arrived on this planet, and I wasn't reading the Astrophysical Journal yet, or anything else for that matter. But yes, I know the rough history. It took some 25 to 35 years to understand the dynamical evolution of a star cluster, modeled as a collection of equal-mass point particles, depending on how you count, since the earliest N-body calculations were performed on modern computers, around 1960. But you know the details better than I do. I guess it is your turn to do some summarizing.

Bob: During the sixties, simulations by Aarseth, Wielen, and others showed how three-body interactions form the dynamical engine at the heart of an N-body system, the agents of change in their evolution. Even if no binary stars were present initially, they would be formed dynamically in simultaneous three-body encounters. And once there, encounters between these binaries and single stars would complicate the normally weak heat flow mediated by two-body relaxation: a single scattering encounter can suddenly release a large amount of energy when a binary increases its internal binding energy by a significant amount.

During the seventies, Henon and Spitzer and co-workers used approximate Monte Carlo Fokker-Planck simulations to model the contraction of the core of a star cluster, on a time scale that is only an order of magnitude larger than that of the half-mass relaxation time scale. This so-called gravothermal catastrophe was predicted in the sixties, and actually observed by Henon in rudimentary form in the sixties as well. But it was seen much more clearly in the seventies in statistical simulations. While more difficult to observe in actual N-body calculations, because of the still low number of particles, a few hundred at most, they were seen there as well. In fact, one could even argue that in the late sixties, some N-body simulations already showed hints of this effect.

In the eighties, finally the behavior of an equal-mass point particle system after core collapse was elucidated, first by Sugimoto and Bettwieser, who showed that a post-collapse cluster can undergo so-called gravothermal oscillations, a series of local mini collapses and expansions in the very center of a star cluster.

Alice: And then Goodman gave a detailed stability analysis in which he predicted the minimal N value for which an equal-mass N-body system would show such behavior. At that point, theoretically the equal-mass evolution was well understood, wouldn't you say?

Bob: I only consider something understood if it comes out of my simulations, in a repeatable and robust way. I have seen too many semi-analytic predictions come and go in my young life to have too much trust in those!

Alice: What Goodman did was finding roots in the complex plane of relatively straightforward and certainly well defined equations; I wouldn't call that semi-analytic, and I certainly would trust the complex plane a lot more than the complexities of any complicated simulation.

Bob: I guess we're talking about matters of taste here, though many would think this an odd thing to say about `hard' science. But moving right along, let me make my main point.

It would be another decade before the predictions would be tested in simulations. In the late eighties, various simulations based on Fokker-Planck approximations as well as gas models verified what Sugimoto and Bettwieser had seen in the early eighties, and in the mid nineties saw the first observation of gravothermal oscillations in a real N-body simulation, by Makino. In a way, this was the end of a chapter in stellar dynamics, and further progress had to come from more realistic systems.

Alice: But surely people had used a mass spectrum long before that.

Bob: Yes, in fact the very first paper by Aarseth already described a multi-mass simulation, in the early sixties. But it was only in the seventies that more detailed studies elucidated the main physical mechanisms of mass segregation.

By the way, the field of stellar dynamics owes a lot to the inspiration provided by Sverre Aarseth. For the last several decades he has shared his codes with anyone interested, operating in an `open source' mode long before the term was invented. Not only that, with his constant readiness to help anyone interested in using his code, he has set the tone for collaboration for at least two generations of stellar dynamicists. I believe that the attitude toward collaboration is more prevalent in stellar dynamics than in most areas in astrophysics, and as such this simple fact may be Sverre's single-handed accomplishment.
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