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3. A Kitchen Sink Approach

3.1. The Need for Combining Codes

Alice: But as long as only a few strange stars appear in a whole cluster, why would you make your cluster evolution code so enormously more complicated in order to try to handle those? Surely the vast majority of stars in any globular cluster will never hit another stars.

Bob: But I am not only interested in the vast majority of the stars. I would like to know the nature of X-ray sources and binaries containing millisecond pulsars. By studying the properties of neutron stars and black holes in binaries, we can learn a lot of fundamental physics. We can learn about dense states of matter in a way we cannot possibly produce experimentally within a laboratory, and we can learn about general relativity in strong fields, right up to the horizon of a black hole. And since black holes are black when nothing falls into them, we have little choice but studying them in a binary context.

Alice: I am certainly interested in these objects as well. But wouldn't it be good enough to first neglect the stellar evolution complications, and later make up for them? As I suggested earlier, you can use a method of successive approximations. First you model the whole cluster history, and only then you take care of the special cases, given that you already know the state of the environment they are in.

Let me give an example. When you calculate the evolution of a single star, it is possible to calculate the evolutionary history using only a rather small network of nuclear reactions. Then, if you are interested in the amount of trace elements formed during the evolution, you can go over the whole history once again, in a second pass. You then take the time dependence of the structure of the star as given by the first pass, and against this background you compute the behavior of a much more detailed nuclear reaction network.

Bob: That wouldn't work here. While it is true that the total number of weird stars, affected by collisions, is small compared to the total number of stars, it may not be small compared to the number of stars in the core. The densest cores in globular clusters contain only about 1 percent of the total mass of a cluster, if that much. And a large fraction of stars in such a core may have undergone a close stellar encounter during its lifetime.

So for the overall cluster properties, you may or may not be right, that's an arguable point. But for the detailed core properties, you are certainly not right. How the merger products behave in the core affects the core parameters, and by changing the local environment, all stars in the core are affected.

Alice: Well, maybe that is true. And, come to think of it, there are other dense stellar systems where encounters are more dominant. If we look at star-forming regions, it now seems pretty clear that many if not most stars are produced by interactions between two or more protostellar gas clouds. So for those systems, you have no choice but combining hydrodynamics and stellar dynamics and stellar evolution. And if we look at galactic nuclei, we quickly realize that the stars in the immediate neighborhood of a central massive black hole have a mean time between collisions that is far shorter than the age of the universe.

So, yes, while I still don't like the idea of a kitchen sink approach, I must admit that there are certainly situations where one has no choice but to go that route. And the outcome of such calculations apply to the formation of all stars, and the final stages of the most interesting end products of stellar evolution. Okay, I'm convinced now!

Bob: Glad to hear that! And of course, I am not advertising a kitchen sink approach for all stars simultaneously. There is plenty of room for stellar evolution tracks and binary evolution recipes, as we saw. But that in itself is good and bad. It is good in that it is easier to check the accuracy of tracks and recipes than the accuracy of the outcome of kitchen sink combinations of codes. But it is bad in the sense that the software package you wind up with is maximally complicated.

Compare the situation with a mindless approach in which you do every stellar evolution from scratch, without using stellar evolution tracks. In addition, you could compute the behavior of each binary by letting two live evolution codes talk to each other, dump mass from the one onto the other if need be, and adjust the orbital parameters according to the mass and energy and angular momentum exchanged. It would cost a lot more computer time, perhaps, but it would certainly make the code simpler.

In contrast, what I have in mind is a code that combines everything: stellar dynamics, stellar hydrodynamics, stellar evolution tracks, recipes for binary star evolution, a live stellar evolution code, and yes, a live binary star evolution code as well. I'm not looking forward to putting all that together, but frankly, I see no other choice.

3.2. Parametrization

Alice: If you really succeed in putting together such a code, and you start doing simulations with it, then what?

Bob: publish the results.

Alice: I suppose so. But what have you learned from it? We have agreed now that such calculations should be done, but it is a different question how to interpret them.

Bob: I don't see what the problem is. After you do a detailed simulation, you just make a reasonable choice of what figures to publish, based on the most relevant physical parameters, and then you write the text around it. And for people who want to know more details, you can make the raw data available. What else can you do with the results of a simulation?

Alice: I mean more than that. You told me that you are interested in learning more about the properties of neutron stars and black holes. A theoretical physicist staring at your published figures of Lagrangian radii is not going to find much deep insight in those. In the end you want to come up with hard numbers describing some of the properties of these exotic objects.

Now in order to do that, you have to find a way to characterize the environment of these objects, at the very least the cluster core, as you stressed already. Since all stars interact with each other in the core, we can only believe the details of a simulation if we can believe the details of all physical processes that play a role in the environment.

Bob: But we have already seen that, regrettably, it will be a very long time indeed, before we will be able to give a quantitative treatment of common envelope evolution and similar such ill understood stellar evolution phases. I think we just have to live with what we can do.

Alice: I think we can do better. Let us take the example of the evolution of a single star. Imagine that you were the first person ever to attempt to do a computer calculation modeling the evolution of a single star. You want to write a computer code, but first you have to decide how to represent the physics. Now I walk by your office, ask what you are going to work on, and then I tell you that it cannot be done, since we don't know convection. Sure, we can make order of magnitude estimates, a form of dimensional analysis. We can talk about the typical size of a convective cell, but beyond that we know next to nothing. How can we possibly make quantitative calculations about a star in which convection plays an important role, if we don't even know the value of the most significant digit describing the local state of a fluid in convective motion?

Bob: I imagine that I would be unhappy, hearing that, but that I would argue that I would try anyway, just to see how good or bad the outcome of my calculation would be. After all, that is what happened historically, and we're all glad people didn't give up in the late fifties!

Alice: Well, yes, that is what happened, but there was an essential step that was taken first. Even without any detailed knowledge of convection, a brave step was taken by introducing the concept of convective scale length h. And while there really was no reason to believe that this notion had much physical meaning beyond an order of magnitude characterization, it did provide us with a quantitative handle, a number h that could be determined afterward, by comparing theory and observation.

It was altogether possible that no single good value for h would have reproduced the observed structure and evolution of stars. Perhaps convection should have been modeled by a function that is fully dependent on temperature and density and chemical composition. A priori you just don't know. The happy surprise was that a single number h turned out to be enough to find rather good agreement between the model produced and the stars observed.

What I would propose for a kitchen sink calculation of a globular cluster, or any dense stellar system for that matter, would be a parametrization of all the unknown physics, by however many parameters it would take.

3.3. Comparison with Observation

Bob: You'll wind up with dozens of parameters, do you realize that?

Alice: I do, and yes, that may present us with significant problems. But now it is my turn to say: what else can we do? But I can say more. First of all, any globular cluster simulation already requires you to choose dozens of parameters. There is the choice of initial model for the density distribution, if we assume the simplest case of isotropy. Then there is the choice of mass spectrum. And for the primordial binaries you have to give a prescription for the initial distribution function, as a function of mass ratios, separation and eccentricity.

Bob: And a prescription for the tidal field, which can be time dependent. And perhaps you'd like to take into account disk shocking, when the cluster passes through the galactic plane. Yes, I know, there are many numbers you have to specify already. And these may well be different for different globular clusters.

Alice: Exactly, that is my point. If you can make detailed realistic models for a number of different globular clusters, you have to provide all those initial conditions, and vary them until you get good fits with the observations. So the situation will not be much worse if you add a handful of parameters modeling the physical uncertainties. And the good thing is, a parameter describing the outcome of common envelope evolution, for example, can be expected to be the same for different clusters, at least in first approximation. So the more clusters you model, the less these extra parameters will cause problems.

Even so, it still seems worrisome to fit dozens of parameters to one set of observations. There is this nice quote about John von Neumann, who certainly knew how to use computers to fit data. He used to say: "with four parameters I can fit an elephant, and with five I can make him wiggle his trunk."

Bob: It just shows that globular clusters are more complicated than elephants.

Alice: I don't think von Neumann would have agreed. But it all depends on the accuracy of the available data, and most importantly, upon the lack of degeneracy in the types of data available. And this brings me to my second point. If we would just try to fit density profiles and velocity dispersions, we would be hopelessly lost. The only thing that can save us is detailed observations of specific types of stars, such as binaries of different types. In that way, we can hope to obtain more independent constraints than there are independent initial conditions.

Bob: Are you thinking of X-ray binaries and binary pulsars? Given the many hundreds of such X-ray sources that have been found in galactic globular clusters, they can help constrain quite a few parameters already. And similarly, the wealth of pulsars that is being discovered in globulars is constantly growing as well.

Alice: Yes, but I am also thinking about more normal binaries. If future observations can pin down more of the parameters of the current population of binaries, we still will have to search around in parameter space to find the original distribution, but at least that will constrain the uncertainty quite a bit. Primordial binaries provide the largest number of free parameters in setting up a realistic set of initial conditions for a cluster evolution, I bet.

Bob: That is right. And yes, the observations are improving. Also, I don't expect to have a working kitchen sink code any time soon. I fully trust the observers to have quite a bit more data in hand by the time my code will work and is debugged.

Alice: Is a code ever fully debugged?

Bob: Well, good enough is good enough, I'd say. No person is perfect, and no code is perfect.

Alice: Given that you have convinced me that a kitchen sink code is a reasonable goal to work toward, I'm curious to hear a bit more about how you intend to write such a code.
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