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9. Standard Units

9.1. Confusion

Alice: Now that we have a working and well-tested code to generate realizations of Plummer's model, we have to decide how to scale the output. We simply started with the coordinate system that was used by Aarseth and his friends, in which . While that is a reasonable choice, there are other choices as well.

The problem is: if we make one choice in one program, and another choice in another program, we are inviting disaster. If you then put those programs together, as different modules in a single larger context, you will generate nonsense.

Bob: Well, let's make a particular choice then, and stick to it.

Alice: What have you been using so far?

Bob: I have never settled on one particular choice. It always depended on the problem at hand, the code I used, and the preferences of my collaborators.

Alice: People always underestimate the importance of standardization. It takes some work, of course, for a community to settle on a standard. However, it takes far more work if you always have to transform between different systems, even in the unlikely case that you do not make mistakes. Scaling the results from one system to the others is tricky.

Bob: Tell me about it! Or better, tell my students about it. One thing students always have great trouble with is scaling the results from an N-body simulation back to physical quantities, expressed in physical or astrophysical units.

Alice: That is not surprising. And I must say, I often have to think carefully about such questions. It may be trivial, from a scientific point of view, but in practice, it is easy to make a mistake. It is one thing to get as an answer that your star cluster retains a final mass of 0.765 at the end of a run, but it is quite another to translate that into grams, or into solar masses.

Bob: At least the physical units are standardized. Mass comes in grams, or in kilograms. It is annoying that there are still two sets of units in general use in astrophysics, MKS and cgs, but at least the conversion between those two is relatively straightforward, just a factor of a thousand in the case of masses. And similarly, astrophysical units are standardized: mass general comes in units of a solar mass. But in computer simulations, everybody uses whatever convention they like.

I often find it difficult to interpret the results that come from running someone else's code. Not everybody clearly documents what units they used. I know, I know -- I just told you that I have never settled on a single system of units either. I wish there was such a system.

9.2. A Standard

Alice: But there is a standard for computer simulations in stellar dynamics. And I have been using that standard ever since I started running simulations.

Bob: What are they called?

Alice: Hmm. They are generally referred to as `standard units.' Perhaps that is one reason that they haven't found general acceptance yet. Maybe we should give them a real name! They are sometimes also referred to as `Heggie units,' since Douglas Heggie was the first one, as far as I know, to stress the need for such standardization.

Bob: Did he publish the definition of his standards?

Alice: Yes, in a paper with Bob Mathieu, back in 1986, as a contribution to a conference where this issue of standardization was discussed. The reference is Standardised Units and Time Scales, by Douglas Heggie and Robert Mathieu, and it appeared in 1986, on page 233 in The Use of Supercomputers in Stellar Dynamics, edited by Steve McMillan and Piet Hut, and published by Springer.

Bob: How did they define their units?

Alice: They took the gravitational constant and total mass of a star cluster to be unity, and they took the total energy to be -1/4.

Bob: Why one quarter? Unlike the first two choices, that doesn't sound very natural to me. Why not unity?

Alice: Actually, the origin of their definition stems from the fact that they took the virial radius to be unity. I should have introduced their choice as:

The fact that is a consequence from this fact.

Bob: I know the virial theorem, but what is a virial radius?

Alice: For an equal mass system, an elegant definition is: the virial radius is the inverse of the average inverse distance between particles in an N-body system. Expressed as a formula:

averaged over all particle pairs . For a general system of particles with masses and total mass M, the definition is:

and of course, we recognize that this is the potential energy of an N-body system, for , apart from a factor of two, because we have now counted every pair twice. So we can write the potential energy of an N-body system as:

Bob: And now we can use the virial theorem, which tells us that the magnitude of the total energy is half that of the potential energy, to write:

and with our previous choice of , we get:

Now I see what you meant, with the total energy of -1/4 being a consequence the choice of . But I'm still puzzled. What is so special about the virial radius that you want to set it equal to unity? Why not set the total energy equal to unity, say?

9.3. Motivation

Alice: I think that the original idea was that we would like to choose the most natural units for the three basic physical units, namely those of mass, length, and time. But since we also like to scale the gravitational constant to unity, we have only two degrees of freedom left over. The total mass is an obvious candidate, since it appears in many equations as , which is nice to forget about by equating it to . It also means that for equal-mass N-body systems, you can count on each particle always having a mass of .

The only remaining question is: what to do with the last degree of freedom. Do you want to find a natural length scale or a natural time scale, or do you want to take a more derived quantity, not directly coupled to the basic physical units, such as the energy? I for one agree with Heggie and Mathieu that it is more elegant to choose a basic physical quantity, either a length or time scale.

Bob: I think I would prefer total energy, derived or not. But if you insist on purity, well, a natural length scale for a star cluster is the half-mass radius . And a natural time scale in turn is the crossing time at the half-mass radius, the typical time for a particle to cross the system, starting at the half-mass radius.

Alice: It is already clear from your suggestion that choosing a length scale is somewhat more natural than choosing a time scale, since in your time scale definition you make use of an earlier length scale definition.

Bob: So what is wrong with the choice of ?

Alice: There is nothing wrong with that, and people have used that choice as well. The problem is that in that case the total energy has a non-obvious value, typically somewhere like , but not exactly.

Bob: Well, my original suggestion was to make . Not only is that an exact number, it is a very simple number. I'm still not sure what is wrong with that.

Alice: If you choose the absolute value of the total energy to be one, your half-mass radius comes out to be very small, about 1/5, and that is not a very natural value.

Bob: Hmm, yes, it would be a bit of a nuisance, to deal with a core radius of , say, and then having to remember that we are dealing with a not very concentrated cluster, since and therefore .

Alice: To sum up, we really would like to have a system of length units, in which the half-mass radius is close to unity. However, the half-mass radius is not a conserved quantity, and as soon as you start a simulation, the half-mass radius may change. Therefore, it is better to take a conserved quantity, such as the total energy, as a gauge, and give it a simple value in such a way that it implies that the unit of length is at least close to the half-mass radius. This must have been the sort of thinking that went into the definition of the standard units, I'm pretty sure.

Bob: That all makes sense. But in practice, the half-mass radius does not change much, if you simulate a star cluster. Only after core collapse does the half-mass radius begin to expand.

Alice: You are used to dealing with a system that starts in dynamic equilibrium. However, if you start with a cold collapse, or a system that has too much kinetic energy and starts off expanding, in both cases the half-mass radius will change right away, while the total energy will remain conserved.

Bob: Okay, I see the advantages of the standard units. And since I don't feel very strong about my other two candidates for standardization, I'm happy to use those virial units, what did you call them, Heggie units?

Alice: Yes, were it not for the fact that Douglas Heggie is a modest gentleman, who would be the first to point out that those units have been used by others before he suggested them. Virial units might actually be a reasonable name; I haven't heard that expression yet.

Bob: Hmmm. It just slipped out, but to me it sounds too much like a medical term, reminding me of a virus. And I certainly don't like computer viruses. I prefer the term Heggie units: he should get credit for his suggestion.

Alice: We'll see what the field decides.

9.4. Approximations

Bob: By the way, I was impressed by the fact that you juggled those numbers so easily, like that value 0.2 that you pulled out of a hat. What was that again?

Alice: That was the value for the half-mass radius, if you would insist on a total mass of unity.

Bob: Ah yes, did you make that up to impress me, or did you calculate or guestimate that quickly?

Alice: None of the above. Since I have been working with these standard units for a long time, and especially since I have been teaching it to my students, some of these numbers just stick in my mind. You mentioned from the start that students always have problems with scaling, and my students are no exception.

Bob: I guess the counter-intuitive aspect is that if your ruler shrinks, everything you measure becomes bigger, and similarly, if you take a ruler with larger units of length, the whole world gets smaller, in terms of the values you read off. Knowing where to multiply and where to divide is something that requires some thought. With one ruler changing, you have to be careful, and if you simultaneously change your ruler, your clock, and your scales, changing your units of length, time, and mass, it is real easy to go wrong.

Alice: I know from experience! And still, I always have to double check.

Bob: Glad to hear we share the same problem. And just to make sure that I can buy into your story, shall we quickly check with Plummer's model as a concrete application how we can derive the numbers you mentioned?

Alice: Good idea! It never hurts to check, as we've now seen a number of times. But I don't like working with factors like and . Let's make some simplifications, trying to use only fractions like and and the like, but let's not get more accurate. That should be enough to show our main point.

So let us start with the original expressions for the total energy which we may as well abbreviate as , the half-mass radius and the virial radius , given in terms of the structural length . Remember that was what we first encountered as the softening length, when we smoothed the potential of point particles. We started off with our Plummer potential

We also computed the potential energy of Plummer's model in eq. (
93). The virial theorem tells us that the total energy is just half that value:

In terms of , we can write, with :

The expression for the half mass radius we already determined earlier, in (100), and the last result follows directly from eq. (110), which tells us that the virial radius is .

I have split off the factor in the expression for the energy, so that we can deal with remaining numbers that are all of order unity, in the form of fractions of small integers.

All we have to do now is to choose different values for . For each choice of , we can see explicitly how everything else will receive different values. The structural length plays the role of our ruler.

9.5. Three Round Numbers

Bob: That is a nice way to lay it all out. Okay, so we have talked about three choices of units, based on what we choose to set equal to one: the energy, the half-mass radius, and the virial radius. Let us start with the energy

Alice: If we take , we are forced to take which implies and .

Bob: How simple! Yes, it is clear now, and you were right about that value of roughly for the half-mass radius. I'm curious to see what the other two choices will lead to. Let us make a list:

Alice: Yes, that is a good summary, and it is important to indicate which relations are approximate and which are exact because they follow from the definitions.

Bob: And these are useful numbers to remember. Actually, now that we have decided to adopt the `standard units' as standard units (we have to come up with a better name), it is only the bottom line that is really worth remembering.

Alice: And since for all models in the standard units, for Plummer's model there is only one number to remember: the fact that the half-mass radius is roughly .

Bob: Ah, but there is also the structural length. Let's see, in standard units that is roughly . That is a second useful number to remember, since it gives a measure for the size of the core of the potential. So we have:

And this shows that Plummer's model is not very centrally condensed: the core is barely smaller than the half-mass radius.

Alice: Talking about central concentrations, it would be nice to throw in the core radius as well, for good measure.

9.6. Surface Density

Bob: The problem with the core radius is that there are several definitions. Which one do we choose?

Alice: There may be several, but the definition that I have seen used most often is the one that appeals to observers: it defines the core radius as the distance from the center on the sky where the projected light density has dropped by a factor of two, with respect to the central value. In other words

Bob: It may be useful for observers, but for a theorist it is rather unnatural to do a line-of-sight integration through a model. But I too have come across this definition quite often, so let's adopt it.

Alice: We'll have to do the integral of course. Better first to draw a picture.

[We should put a picture here]

               / |
            r /  | z
             /   |
            /    |
     - - - o- - -+- - - - - -
              d  |

From this figure it is clear that the surface density, at projected distance from the center, is given by:

where I have used eq. (
11) for the density.

Bob: Now before you will begin to solve this with pen and paper, let me give you the answer, by using a symbolic manipulation program. Here it is: the answer for the definite integral is .

Alice: The impatience of youth! But okay, I'll use your value. We then have:

Bob: And if you want to see the complete dependence on the original variables, dimensional analysis shows that:

where I have switched back to notation.

9.7. More Round Numbers

Alice: We're almost there: the core radius is defined as the place where the surface density has dropped by a factor of two:

So this is the value for the core radius.

Bob: And that gives us a third number to remember for Plummer's model:

Alice: I like that! An unlikely simple progression. I thought I knew Plummer's model by now, but I had not realized how simple the ratios for these main numbers are. Okay, let's remember them all!

Bob: Or, to be a bit more lazy, I may just remember that the half-mass radius is twice as large as the core radius, that the structural radius lies about half-way in between, and that the virial radius is a bit larger than the half-mass radius.

Alice: If we want to be really lazy, we should group all these results together, so that we can later easily come back to look them up:

Actually, what we really need is to know what these quantities look like in standard units:

In the last column we recognize those nice round numbers we just discovered. And while we're at it, we may as well throw in the first and third quartile radii, which we determined before, in eqs. (99) and (101):

Bob: That is a very useful collection of numbers. For example, it tells us immediately that the core of Plummer's model contains a little less than one quarter of the total mass.

Also, if you are using softened particles, you may want to know how extended the mass distribution is. These relations tell you that within one softening length there is just a bit more than a quarter of the total mass, but that within two softening lengths you already have almost three quarters of the total mass.

Alice: And if we agree to stick to these standard units, let us adapt your code, so that it generates Plummer model realizations with the right units.

Bob: Okay, that should be easy.
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