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2. Stellar Evolution

2.1. The Role of Stellar Evolution

Alice: I guess mass segregation had already been observed in equal-mass systems, as soon as hard binaries were formed, since most of the time these tight binaries behave as point masses that have twice as much mass as single stars.

Bob: Yes and no. If you would neglect three-body interactions, you would be right. And indeed, if you would start a simulation with a significant number of primordial binaries, that would definitely be the case. But those simulations were done later, for the first time in the late seventies, and more systematically in the early nineties, after observations had shown that globular clusters have indeed significant numbers of primordial binaries. In contrast, if you only deal with dynamically formed binaries, you don't see much subsequent mass segregation. First of all, such binaries are formed in the core, where the density is highest. Secondly, recoil from three-body reactions will tend to kick binaries away from the center, even while mass segregation will tend to let them move back toward the inner part of the core.

Alice: I see. So mass segregation was studied systematically only in the seventies?

Bob: Yes. Monte Carlo Fokker Planck simulations as well as direct N-body integrations showed in detail how heavier stars tend to wind up in or near the core of a star cluster. At the same time, lighter stars were seen to be more likely to escape from the system. However, the problem with these early multi-mass calculations is that they were inconsistent.

Alice: Inconsistent?

Bob: They were not very realistic, by and large, since most of them did not include the effects of stellar evolution. In the real world, while heavier stars do tend to sink to the center of a star cluster, roughly on a two-body relaxation time scale, they also tend to burn up more quickly. As you know, a star 25 times more massive than the sun burns up a 1000 times faster. So it would be totally unrealistic to follow a multi-mass system, and to leave the heavier mass particles in the center, just sitting there. You really would have to remove them, or at least most of their mass, as soon as their age exceeds that dictated by stellar evolution. Few multi-mass simulations did that.

There were some exceptions. In the early eighties Aarseth and coworkers in their simulations began to remove some or all of the mass of stars after they reached the end of their life. But most multi-mass simulations in the seventies and eighties did not include stellar evolution.

Alice: But it can't be that hard to just diminish the mass of a point particle at the end of its nominal lifetime, from that of a main sequence star to that of a white dwarf, neutron star, or black hole, as the case may be.

Bob: Or put it equal to zero, for those stars where you believe that no remnant is left at all. Yes, that can be done, but that will not get you very far. Fiddling only with the masses of stars is not enough. You would then be faced with the question: what to do with binary stars? Take two stars in a binary with a semi-major axis that is much larger than the sum of their radii while they are on the main sequence. Soon before the heaviest star will end its life, it will want to evolve into a giant phase, and as a result, it may well dump much of its mass on its companion star. How are you going to model that? Not by simply removing the mass from the system. That would not do justice to a large fraction of close binary evolution -- and close binaries are exactly the dynamical heat source of an N-body system.

In other words, the only way to make a meaningful simulation of a non-equal-mass system of particles is to add a realistic form of stellar evolution, for single stars and especially for binary stars. Nothing else makes sense to compare with observations.

2.2. The Case for Including Stellar Evolution

Alice: You like to make strong statements, but I can make a few too! We don't know much of anything about the quantitative evolution of close binaries, so what is the point of adding a largely unknown quantity to an otherwise very detailed stellar dynamical situation?

But let me answer my own question, if I may. We know that the dynamical evolution of a star cluster is not very sensitive to the nature of the central heat source. As long as we know the amount of heat lost at the half-mass radius, the core will adjust itself to whatever size it needs to have to let the central heat source produce the amount of energy needed. The situation is quite similar to the evolution of a single star, such as the sun: given the heat lost from the photosphere, the central temperature and density can adjust quite easily to create the right amount of energy in nuclear reactions.

Bob: Well, that may give you a rough agreement for the overall structure, but for me that is not good enough. I am interested in predicting the number of X-ray binaries and millisecond pulsars and their binaries, as well as the physical characteristics of those systems. They are important from an observational point of view. As you know, a significant fraction of X-ray binaries are located in globular clusters, even though these clusters contain much less than 1 percent of all the stars in our galaxy. It is clear that the vast majority of those X-ray binaries in clusters are formed through dynamical effects, as a result of the high density of stars.

Alice: But you can do that in a two-step process. First you make a good enough dynamical model of the dynamical evolution of a star cluster. Since X-ray binaries and binary pulsars form only trace populations, neglecting them will not greatly influence the overall dynamical evolution, I would think. Now, having done that first step, you can then statistically sprinkle your exotic binaries into the outcome of your first step. This second step will have a Monte Carlo nature, but who cares? Your precise N-body calculation started of from random initial conditions, after all.

Bob: Not so fast. Have you ever looked at the HR diagrams based on recent observations of globular clusters?

Alice: Not recently, no. I do remember that astronomers traditionally talked about something like a second parameter effect. What was that again. I believe that metallicity was the first parameter, and that it was claimed that you need at least one more parameter. In other words, two clusters of the same metallicity can still show different types of HR diagrams.

Bob: Yes, that was a long time ago, even before the first observations with the Hubble Space Telescope. Since then we have learned that there is an amazing variety of morphologies in the HR diagrams. And here we are talking about large numbers of stars, not your occasional X-ray source or other exotic object. The only way we are ever going to explain the structure and evolution of globular clusters is by making our simulations detailed enough to at least be able to reproduce, say, the position and shape of the horizontal branch in each well observed globular cluster.

Alice: Okay, you make a strong case for including some form of stellar evolution right away, when performing a star cluster simulation, besides the basic stellar dynamics. The main question then is: how much? What is wrong with using a rather crude toy model?

Bob: I can't imagine modeling globular clusters on a cluster by cluster basis using such a rough approach. I'm afraid we have to do better than that. At the same time, I know that there are large uncertainties in our understanding of contact binary evolution. But what can we do? We do the best we can, and hopefully, `the best' will get better every year, perhaps when our stellar evolution colleagues will use two- and three-dimensional models.

Alice: I won't hold my breath. But just to set the stage, why don't you continue your historical summary. When did people start using more detailed stellar evolution information in their dynamical simulations of globular clusters?

2.3. Tracks and Recipes for (Binary) Star Evolution

Bob: It was only in the nineties that a few groups began to use look-up tables for single star stellar evolution, and simple recipes to describe what would happen to the evolution of binary stars. While they followed the stellar dynamical evolution in the point mass approximation, as before, their stars would loose mass, and their binary stars would exchange mass, in a roughly realistic fashion.

Actually, it is pretty amazing that it took so long for stellar dynamics and stellar evolution to come together, even in this rather rudimentary way. It could easily have been done twenty years earlier. After all, in the early seventies, single star evolution was well understood, at least in outline, and plenty of evolution tracks had become available. And the main phases of binary star evolution, including the conditions for gradual and catastrophic mass transfer, had been analyzed as well.

Alice: Yes, I remember learning about cases A, B, and C of mass transfer in binary stars, during the undergraduate course in stellar evolution that followed. Why do you think people waited twenty years?

Bob: I don't quite know. But if I were to guess, I think the problem is that the bridge was made from stellar dynamics to stellar evolution. Stellar dynamicists had enough problems on their hands as it was, to figure out what happened during core collapse and afterward, with gravothermal oscillations. Only after all that had been sorted out, and especially with the observational discovery of primordial binaries, did an urgent need for stellar evolution treatments make itself felt.

Alice: But you had just convinced me that mass segregation simulations were unrealistic without stellar evolution. This would suggest that Aarseth should have added stellar evolution to his code in the early sixties!

Bob: Well, he was the first one to do so, as I mentioned, but I guess he too had more urgent problems to deal with early on, such as the treatment of close binaries, inventing and adapting regularization schemes to avoid numerical errors during close encounters. But in the end, I think astrophysicists have inertia, like all human beings: stellar dynamicists just won't get excited about bringing in stellar evolution in their codes until they really have to.

Alice: So why did stellar evolution folks not add dynamics to their simulations, or at least knock on the door of dynamicists?

Bob: They, too, could have and perhaps should have done so in the seventies. They did start to do population synthesis studies around that time, Tinsley and others. But of course, they too had other worries to take care of first. It was already a big job to get single star evolution treated correctly in a statistical fashion, for a complete star cluster or a whole galaxy. Then, and even now, there are significant questions about what exactly happens to various types of stars in the late phases of their evolution. And when people started to add the signatures of binary star evolution, they quickly realized how little we know about many important phases of close binary evolution, and contact binaries in particular.

So it is perhaps not too surprising that they did not rush to put in dynamical effects of stellar collisions. Even though blue stragglers had been known to exist in star clusters since the fifties, it was not yet clear how important collisions were in producing those stragglers, notwithstanding the pioneering work by Hills in the seventies. In fact, the first conference dedicated to blue stragglers was held in the early nineties, and the first conference on stellar collisions took place in the year 2000!

In contrast, tidal capture of a neutron star by a main sequence had been studied in the mid seventies, because it seemed to be a promising route to explaining the large overabundance of X-ray binaries in globular clusters, which I mentioned before. But that type of analysis was largely local, studying individual encounters, rather than modeling the behavior of a whole cluster in much detail.

Alice: Given that it took twenty years for stellar dynamics and stellar evolution to meet, in evolution recipe mode, once they met it must have produced a major shift in star cluster modeling. Before that, most simulations were exercises in mathematical physics, a lot of fun in itself, but of limited use for comparison with observations, as you have just argued. But once you put in tracks and recipes, aren't you in a good position to start explaining the observational data?

Bob: No, not really. I consider the stage of using recipes as not much more than playing scales, as warm-up exercises before moving on to the type of kitchen sink simulations that I mentioned before.

2.4. Limitations of Tracks and Recipes

Alice: Let me come back to my previous objection. You admitted that we know rather little about the evolution of contact binaries. What is the point of computing detailed stellar evolution models if there are vast areas of stellar evolution where we don't even have a clue of what is going on, beyond a qualitative hand-waving hunch? Take the case of common envelope evolution. If a white dwarf starts to spiral in toward the core of a red giant, it seems plausible from an energetic argument that the envelope of the giant might be lost before the white dwarf reaches the core, leaving behind a tight pair of what will look like two white dwarfs. And it seems equally plausible that not enough mass is lost not fast enough, and that the white dwarf will merge with the core. Detailed 3D calculations of this process are very hard to do, given the fact that the initial stages cover very many dynamical crossing times.

Given this fundamental uncertainty, why bother doing detailed calculations elsewhere in a simulation? A chain is as weak as its weakest link. It seems like a waste of computer time and software effort to build some very strong and detailed links as long as other links such as common envelope evolution are too weak to contemplate.

Bob: To some extent I agree with you. First of all, okay, I see no use for live stellar evolution codes to compute the evolution of single stars, within a stellar dynamical simulation. And as a second okay, I agree that even the evolution of primordial binaries can be treated adequately through a combination of recipes and stellar evolution tracks. Where I differ from you is in my view of the treatment of merger products.

Alice: Before we get to differences, let me point out that your first and second okay are very different types of okay. The first one applies to the use of relatively accurate and robust information. There is pretty good agreement between different stellar evolution experts as to the quantitative behavior of the tracks of normal stars, apart from perhaps the very most massive stars. Your second okay addresses the use of rather ad hoc and rough treatments of binary star evolution, where quantitative certainty is far less good, some would say almost absent.

Bob: Yes, I agree with all that, but what can a poor boy do? We do the best we can. And in order to get at least some new insight in the evolution of star clusters, I think those tracks and recipes are good enough, even though the latter are far from ideal, of course. But let me move on to my main difference with your view. Whenever two stars collide with each other and merge, you wind up with a merger product that is totally unlike the type of normal ZAMS (zero age main sequence) star that stellar evolution tracks all start with.

A merger remnant has not only a very different metallicity than ZAMS stars in the same cluster had, what is worse, the chemical composition is different at different radii in the star, due to incomplete mixing during the collision, which in general will be significantly off center. In addition, for a hundred million years or so, the merger product will be out of thermal equilibrium, and therefore will have a quite different structure from a normal star.

The only way you are getting even roughly close to determining the structure of such a star is to use not only a live stellar evolution code, but also a live hydrodynamics code to follow the collision. So what I envision is that when two stars come close together within the stellar dynamics part of the simulation, these point particles are handed over to a hydrodynamics code, which replaces them with blobs of, say, SPH particles, layered in the proper way as specified by the stellar evolution information in the simulation. From this point on, the power of SPH is let loose until we arrive at a dynamically settled merger remnant. A stellar evolution code will then follow the thermal settling, as well as the subsequent more normal evolution.

But if this does not convince you, consider what happens subsequently with a merger remnant. It will be formed in the core of the cluster, most likely, since there the chance for collisions is highest. It will remain in the core, since on average it will be more massive than typical single stars. Therefore, it will stand a significant chance to undergo yet another collision, or be captured as a binary member in an exchange reaction. Even if it avoids collision through such a three-body dance, subsequent evolution is likely to lead to mass overflow. How can you possibly use recipes to treat mass overflow between stars that are not parametrized only by mass and chemical composition, but by the full functional dependence of weird composition gradients, and are possibly still out of thermal equilibrium?

To sum up, while it is possible to make tables of stellar evolution tracks for unperturbed stars, and while it is just possible to combine pairs of these tracks with elaborate prescriptions for any conceivable combination of two stars in orbit around each other, it is utterly impossible to prepare beforehand for all types of strange merger remnants that can be formed, let alone for their subsequent interactions.
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