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## 8.1. Regularization

Bob: As for the stellar dynamics, how fancy shall we make our code?

Alice: At least we should be able to handle core collapse for a small number of particles, say a few hundred or so, so that we can model a small cluster, and see the inner parts shrink gravothermally.

Bob: In that case, it would be nice to be able to deal with close binaries as well. Otherwise we have to stop the simulation before core collapse is completed. In addition, we may want to study the effect of primordial binaries.

Alice: But I would prefer not to get into regularization and all that. Whatever I have seen from special treatment of close encounters convinced me that that whole topic is rather specialized, and not something we want to throw at the students right away. Wouldn't it be sufficient to use only one integration scheme globally, for all particles, without making any local exceptions?

Bob: As for the global dynamics, we are pretty much forced to introduce an individual time step scheme. That will require quite a bit of explaining to the students, since it goes way beyond what they can find in the standard text books on numerically solving differential equations, but we have little choice. If all particles would share the same time step size, the whole system would get down on its knees when somewhere two stars would come close together, or worse, would form a tight binary.

However, even with individual time steps, almost all of the computer time for a few hundred body system could go into just one tight binary. And what is worse, the integration of such a binary could lead to an unacceptable build-up of roundoff errors.

Here is an example. It can easily happen that two stars form a hard binary, with a semi-major axis that is a thousand times smaller than the size of the system. The eccentricity e will fluctuate, due to the perturbing forces from encounters with other stars, and occasionally, e can get close to unity. If, say, e = 0.999, then at the distance of closest approach the stars will be a million times closer than the size of the system. In order to calculate the force between the stars, we have to first subtract the position vectors of both stars, and this will lead to a loss of precision of at least six digits, reducing an original 64-bit double precision calculation almost to 32-bit single precision. We simply cannot afford that, if we want to integrate the system in an accurate way.

One way to overcome this problem is to introduce a special coordinate system. Expressed in relative coordinates, with respect to the center of mass of the two stars, there is no longer any problem of round-off. An even better way is to go one step further: to replace the orbit of a very tight binary star by the analytic solution of the Kepler problem, an elliptical orbit for which we only have to solve Kepler's equation to know where the two particles are with respect to each other, at any given time. This will be an excellent approximation when there are no other stars anywhere in the area.

Now sooner or later, a third star may happen to come near. In that case we can temporarily switch back to numerical integration. Or, if the binary has a very eccentric orbit, we may want to avoid significant numerical errors by introducing a celestial mechanics perturbation treatment. In that case, we stick to an analytical Kepler orbit as the unperturbed solution, and we integrate the perturbation equations. But in general, the best solution is to use direct numerical integration in four dimenstions, using what is called a Kustaanheimo-Stiefel transformation, or other variants that have been introduced later. These of course are only the first steps. There are many other very nice tricks of the trade. If you use chain regularization and Stumpf . . .

Alice: Yes, yes, I'm sure there are many wonderful tricks, but we have to offer your students something that they can understand and apply, all within half a year. Whoever Stumpf is, or the other two gentlemen you mentioned, let's leave them out for now.

Bob: But we should do something, on the local level, to make the whole calculation meaningful. The question is what we can introduce that goes partway toward full regularization. One thing we cannot get around is to do the analytic regularization that I just mentioned. Perhaps that is good enough to get them started. It will actually be instructive for the students to see how things fail, and then to see what solutions can be found to repair the situation. The only way to do that, I think, would be to introduce the separate coordinate patches I talked about. While that will rapidly get too complicated, it would be fun, and it would give students a more realistic sense of the complexities that a real code has to deal with. Who knows, when we really get going, we might even want to give that a try.

Alice: Aha, do I detect the possibility of a smooth evolution from a toy model to an actual production code?

Bob: No no, that would be far more complex. But it would be a step in the right direction, for sure.

Alice: If the main problem would be a loss of accuracy, how about use multiple precision? Instead of 64-bit word length for floating point numbers, using 128-bit word length?

Bob: You mean quadruple precision? Yes, that would be an option, but it will slow you down. Depending on the machine you use, it could slow you down anywhere from a factor of a few to a factor of a hundred or so, if you can do it at all: not every compiler has a quadruple precision option, and you don't want to write all the routines by hand for computing multiplication and division and square roots and all that.

In any case, yes, there are plenty of options. Right now, I wouldn't be able to guess what would and would not work well, under which conditions. So I'll be learning quite a bit from this exercise, I think, even though it's only a toy model. The more I think about it, the more it seems like a fun project.

## 8.2. Local versus Global

Alice: If we really cannot avoid introducing some local treatment in the stellar dynamics part of the code, it would be best to split the stellar dynamics itself into two modules, and a clear interface.

Bob: I told you, that you would want to introduce more modules, before long! How many do we have now? A scheduler (SC), a stellar evolution (SE) and a hydro (SH) module, and now a global stellar dynamics (GD) and a local stellar dynamics (LD) module, five in total!

Alice: Nothing wrong with having five modules in a code. This means that you have a lot of freedom! You can replace each module, rewrite it, experiment with it, all without affecting anything in any of the other modules. But let's not repeat our arguments. After we finish our first toy code, we can take up this discussion again, and by that time we'll have a lot of actual code to work with, to strengthen our arguments.

Bob: Fair enough. But now I'm puzzled. Passing information between the SD and SE module was natural enough, since we were dealing with two rather different types of physics. But the GD and LD modules both are dealing with the same physics, plain old Newtonian gravitational attraction. Putting an interface between them would seem like trying to draw a line on the water!

Alice: I want to draw a line in the sand, frankly, and insist that we divide the stellar dynamics into these two different modules. Look at it this way. From the point of view of the global dynamics, the local dynamics is a way to protect ourselves from the Kepler singularities, right? I presume that is the meaning of the word regularization, removing the singularities.

Bob: Yes, I think that is where the term comes from.

Alice: In that case, the SH module is also a form of regularization. When two point particles come close, the SH module will replace them by, for example, a blob of SPH particles. We could call this a form of hydro regularization. It looks a bit like softening, where every point particle is replaced by a small Plummer model. Softening could also be called regularization.

Strictly speaking, the mathematical term regularization implies that you do not change the underlying equations of motion, so in that sense hydro treatment and softening should not be called regularization. But we are not mathematicians. In astrophysics, we start with extended objects. The idealization of replacing a star by a point mass is only a matter of convenience. And when this replacement turns out to be inconvenient, in close approaches between stars, we are free to use other idealizations. We are only `regularizing' what had been `singularized' as a too extreme idealization.

Bob: But you can't simulate a star cluster with a code that is only using softening. The code will run, but it will not give you a sufficient amount of two-body relaxation. And it will not admit hard binaries.

Alice: That is true if you use a large softening length, like people do when they collide two galaxies, and they want to suppress two-body relaxation, which would be unphysical anyway in that case, since each single particle represents thousands of stars, if not more. But in the case of a star cluster, you could give each star a softening length that is equal to the physical radius of a star. In that case, binaries could form and two-body relaxation will be just fine. You would have to do more, since you would have to prevent stars from passing through each other, as softened particles would. When you thus add a hard core and a friction term to let the particles stick together upon close approach, you have gone well above the simple softening recipe. But if you add those other prescriptions, using a small softening in itself is not necessarily wrong.

Bob: So you view hydro treatment and regularization as somewhat similar, from the point of view of the GD module, which only deals with point particles?

Alice: Yes. And the interface between GD and LD can be similar to what we saw before. We can specify a function to create a local clump of particles, and similarly we can specify a clump destructor function. We can have functions providing the mass and effective radius of a clump, as a function providing the current time. It all follows rather closely to what we just discussed for stellar evolution.

Bob: Hmmm. I still think it is like drawing a line in the water. But early on I agreed to do the experiment, to write a modular toy model, and if this is what you want, this is what we'll try to do. But allow me to laugh loudly if your approach will result in an unwieldy code, with many extra lines to circumvent your interface restrictions, and a very slooooow overall execution speed. And I won't hesitate to tell my students whose brilliant idea it was to do all this ;>).

Alice: If that is what will happen, you can use it as a case study in how not to follow the advice of computer scientists who tell you to use modular programming. I'm convinced that this will not happen, but there is only one way to find out: we will continue, and see who is right.

Bob: I can't wait!

## 8.3. The Name of the Game

Alice: I can't wait either, and whatever the outcome will be, it will be a useful experiment. Before we get started, though, there is one more thing we have to decide. As long as we put all this wonderful stuff soon-to-come on the internet, it would be good to give it a name. I wouldn't want to refer to it as `our kitchen sink toy model' all the time.

Bob: A name. Hmm. Many options. We could use an acronym. Fortran comes from formula translator. How about Dentran, to translate your modular ideas for modeling dense stellar systems?

Alice: That sounds like something to do with dentists, perhaps a new type of tooth paste. How about using a Greek term? The word astrophysics comes from Greek after all.

Bob: I'm not very strong on languages. Do you have a specific suggestion?

Alice: A while ago I asked a Greek astronomer what the word for star cluster research would be. You know, geology is the study of the Earth, astronomy is the study of the stars, so what kind of something-logy or something-nomy would be the study of star clusters? After some thought, he suggested `smenology'. I believe that `smenos' means something like a swarm, like a swarm of bees.

Bob: I am often enough stung by bugs in N-body codes, so that would be quite appropriate, I'm afraid, although when I hear the word bug, I think of small little critters, not as big or ferocious as bees. But how do you turn that into a name?

Alice: We could call it the Smenology Code?

Bob: Too long, too difficult to pronounce, and besides, the logic doesn't work. You wouldn't want to call a code the Astrophysics Code, would you -- as if there would be only one such thing.

Alice: And no comment. Okay, well, what else. You want a short name, I take it?

Bob: The shorter the better.

Alice: The A code?

Bob: Unless you will do all the writing, it should at least be the AB code, for both of our initials. But seriously, it doesn't have to be quite that short. Hey, you like languages; it doesn't have to come from Greek -- why not pick something from Chinese or Sanskrit or some other side street.

Alice: Sanskrit, now that's an idea. How about Kali? That means `dark' in Sanskrit. As in the `kali yuga', the dark ages we are currently in according to Hindu mythology. Or as in Kali, the Goddess who is depicted as black.

Bob: As least the name is conveniently short. And since the universe is by and large a rather dark place, the name is not inappropriate. We probably should include the option of modeling black holes too, with this name. Do you know the Sanskrit for `hole'?

Alice: Beats me. But then again, we wouldn't want to only model holes. And yes, it would be nice if our code would be so robust, simple as it may otherwise be, that it could handle mass ratios of one to a billion, like in a brown dwarf circling around one of the most supermassive black holes in the nucleus of a central galaxy in a rich cluster of galaxies.

Bob: Okay, the Kali code it will be.

## 8.4. An Integration Scheme

Alice: So we have a name for our toy model, and we have decided how we will model the stellar dynamics, the stellar evolution and the stellar hydrodynamics: as simple as possible, but not simpler. Shall we get started?

Bob: Just a moment, we haven't yet decided what to do with the stellar dynamics.

Alice: You suggested an individual time step scheme, and I think that is fine. Actually, we may want to start with shared time steps, but that is a matter of presentation.

Bob: Yes, but the choice of time step is only one choice we have to make. We haven't chosen an integration scheme yet. If you want to start with shared time steps, you can do that using a leapfrog integrator, or more reasonably a fourth-order integrator of one type or another.

Alice: Ah, of course. And yes, plenty of choices. Well, to make it really accessible for students, with no prior background in numerical methods, we should really start with a first-order scheme such as forward-Euler, shouldn't we?

Bob: Really, not with the Hermite scheme?

Alice: What is a Hermite scheme?

Bob: This is the current workhorse of N-body methods in which close encounters can occur.

Alice: I admit that it is a long time ago since I looked at such an N-body code. My recollection was that a rather complicated predictor-corrector method was used, where the force calculation from various previous time steps was remembered. The book keeping was very complicated.

Bob: That must have been a long time ago indeed! For the first thirty years or so, this was indeed the method of choice, until in the late eighties Makino came up with a simpler self-starting scheme, which he called the Hermite integrator, since it uses some ideas put forward first by mister Hermite, a couple centuries ago, I think. This is what is now being used almost exclusively in stellar dynamics of dense stellar systems. And it's not that difficult to code. You can write it in such a way that it looks like an almost obvious generalization of a leapfrog integrator, apart from a few coefficients which would be hard to guess off-hand.

Alice: In that case, it would be good to incorporate that scheme in Kali code. Even so, I like to start simple. I don't want to scare the students by presenting complex integration schemes until they have gotten a sense of what the notion of integration means.

Bob: Wouldn't that be too much hand-holding? Why not let them jump in right away?

Alice: The drawback of giving them a fancy working tool right away would be that they would immediately run with it and move on to fun applications. Chances are they would never bother to think about how and why it can work in the first place. These days, students are all too much focused on learning how to use a package, rather than writing it and getting a feel for what is under the hood.

Bob: I'm afraid you're right. When I did my first programming, it was in basic, and soon afterward in C. Nowadays, when students talk about programming, they mean tweaking the parameters of a package, while using GUIs.

Alice: Perhaps we begin to sound like old folks, shaking our heads about the behavior of youngsters nowadays. But in any case, I really think it is best to start with the simplest possible integration scheme, forward Euler. That way they can get a feel of what it means to follow the tangent of a curve in literally the most straightforward way.

Bob: A clumsy way to start, for sure, but at least they will then learn to appreciate the power of higher-order integration schemes. Okay, let's do that.
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