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1. A Simple Equilibrium Model

1.1. Time Scales

Alice: Hi Bob, I guess we are now getting close to do some real stellar dynamics of star clusters!

Bob: I just had the same thought. We now have a real N-body code, and it is high time that we use it for what it is designed to do: to follow the evolution of a collisional star system.

Alice: I prefer to avoid the term `collisional,' it is just too misleading. Students will think that it implies real physical collisions between stars.

Bob: But it is an old and established term. It won't be easy to avoid it. But I agree, it is misleading. It only means that two-body encounters are important. In other words, two-body relaxation plays an important role in collisional systems, whereas you can neglect such relaxation effects in collisionless systems. In yet other words, in collisional systems, there is a significant heat flow through the system.

Alice: Yes. In stellar evolution they talk about a thermal time scale, like the Kelvin-Helmholtz time scale, a hundred million years or so for a star like the sun. This is the time scale at which a star will lose most of its energy from its surface, if the energy would not be replenished by nuclear reactions in the core of the star. In contrast, the dynamical time scale for a star like the sun is only a couple hours. This is the time it takes a star to `ring', like a drum: the time to cross the star at the speed of sound.

Bob: That is a nice analogy. For a star cluster the dynamical time is called the crossing time. The speed of sound in a star cluster, like in a gas of molecules, is of the order of the speed of the constituent particles.

For a typical globular cluster, with half-mass radius of the order of 10 parsec, and a velocity dispersion of order 10 km/sec, the crossing time is of order a million years -- more than a billion times larger than the dynamical time scale for a typical star.

But the thermal time scale for a globular cluster, the time to redistribute the energy through the collective effects of the diffusion caused by two-body relaxation, is not that much larger than that of the sun: typical values for globular clusters are a billion years, only a factor ten larger than the Sun's Kelvin-Helmholtz time scale.

Alice: Coming back to the term `collisional stellar dynamics', I wonder why gravitational encounters were called `collisions.' One reason may be the analogy with molecular diffusion, where the van der Waals forces between molecules drop off so fast with distance that the only significant encounters are the ones where the molecules practically touch. Also, many of the early simulations of star cluster evolution were Monte Carlo simulations, and there the effects of two-body relaxation are modeled as discrete scattering events, with each star going its own merry way until scattered into a different orbit -- like molecules in a gas.

Bob: Another reason may have been the fact that there was no competition for the term `collisions': it was only in the nineteen-nineties that people had enough computer power to begin to treat collisions between stars in a serious and quantitative way. Back in the sixties, when they talked about collisional stellar dynamics, stars were not supposed to collide, because computers were not up to it yet.

Alice: You may have a point there. In any case, I prefer the term `dense stellar systems' for star systems where encounters are important. Whether the encounters are merely gravitational or also involve occasional physical collisions is less important a distinction.

Bob: Someone else could argue that having collisions or not is the most fundamental distinction. Certainly someone specializing in hydrodynamics is likely to think so. I guess you just betrayed your bias to stellar dynamics. Oh well, classification will also cause heated debates.

Alice: Which I'd rather avoid. Anyway, we're setting out to simulate dense stellar systems, and we know that they have the tendency to show the effects of two-body relaxation: mass segregation, escapers, core collapse, all that good stuff.

Bob: Now that we have an N-body code, let's put up a good show!

1.2. Initial Conditions

Alice: But every show needs a stage that needs to be set up first. What shall we choose for the initial conditions.

Bob: There are many options. Basically, we can just sprinkle particles into a more or less localized region, and in a few crossing times the initial transients will die down. The remaining system, after the fast particles have escaped, will then slowly undergo core collapse.

So we could start with a homogeneous sphere, with a constant density out to a certain radius, and zero density outside that radius. We could give the particles velocities in accord with the virial theorem, or we could even give each star zero velocity: from such a cold collapse, too, you quickly will get a damped remnant, and you will lose less than half of the particles.

Alice: True, but all those solutions are not very elegant. Besides, there is no good reason to pick one over the other. But what is worse, I don't like to mix the transient dynamical effects with the longer-lasting thermal effects. Remember, for a star cluster with a hundred thousand stars there is not much more than a factor of a thousand difference between the crossing time and the two-body relaxation time. And if we want to play with small simulations of only a few hundred stars, the ratio goes down to ten or less. I much prefer to start with a system that is already in dynamic equilibrium to start with.

Bob: Well, we could start with a King model.

Alice: That would be much better, yes, but still we would have to pick a number, such as the central concentration or the depth of the potential well in dimensionless units, in order to settle upon a particular King model, given that there is a one-dimensional family of models.

Bob: What would you prefer?

Alice: How about good old Plummer's model?

Bob: That one? But that's not very realistic!

Alice: At this point I don't care too much about how realistic our simulations will be, if you mean with realistic that the distribution of stars will resemble that of a globular cluster. First of all, Plummer's model does do a reasonable job of fitting some of the more loose clusters, those with a large core radius.

After all, Plummer got his name attached to the Schuster model, in 1911, because he showed that it could be used well to fit the observed cluster data available at that time. While he specifically referred to Schuster's 1883 publication, in which the model was first derived, it is called Plummer's model because of the astrophysical relevance. Sure, we can do better now, but in 1911 Ivan King wasn't even born yet. And you could still travel around in Europe without the need of a passport.

Bob: Those were the days, I suppose. But your choice of N-body simulations were limited to . I'm glad I'm alive now.

Alice: What I like about Plummer's model, in comparison to King Models, is that: 1) it is one well-defined model, rather than a whole family of models, so that if two people simulate Plummer's model, they know that they talk about the exact same model; 2) it is a simple model, where everything can be expressed in terms of analytic expressions, which is not the case for King models; 3) most of the venerable early investigations of star cluster dynamics started with Plummer's Model for the initial conditions, so it is easy to make a connection with the literature.

Bob: From an educational point of view, yes, all three aspects carry some weight. But on the other hand, I always like to introduce students quickly to the more dirty nitty-gritty of actual research. And certainly nowadays you will find far more star cluster simulations starting from King Models than from Plummer's model.

Alice: Okay, let's do both. But if so, it really does make sense to start with Plummer's model. Since everything can be done analytically, the students will get a more direct insight into what is going on. After that, we can move on to King models, and whatever else we find time for.

Bob: Fine with me, since I don't feel as strongly about it as you seem to do. Where shall we start? It would be good to give the students a brief handout with some of the basic facts of Plummer's model. All that I remember about it is that a particle that acquires softening morphs from a point particle with a delta function mass distribution to that of Plummer's model.

Alice: Ah, Great! I knew there was a fourth point I could have mentioned to argue for Plummer's model as a favorite starting point: students are likely to already have encountered it without reflecting on it, when they have implemented, or at least used, particles with a softened potential. Thank you!

Bob: You're welcome, even though you didn't need a point 4), since I had already given in. Okay, let me write down what I remember and what I can easily derive. You, as a champion of Plummer's model, can then add whatever analytical elegance you like.

Alice: Go right ahead!

1.3. A Potential-Density pair

Bob: Okay. A model that is in dynamical equilibrium is defined by a potential-density pair that corresponds to a distribution function in phase space that is time-independent. I would have to scratch my head a bit, in fact quite a bit more than a bit, to remember how you derive the distribution function for a given potential-density pair, to show that an equilibrium solution exists.

Alice: We definitely have to prove that, but let's leave that for later.

Bob: Good! There is only so much I can do from scratch.

Alice: But you have to define what you mean by a distribution function. For a star cluster, it is the density of stars in phase space.

Bob: Yes. Okay, here is the softened potential:

Dr. Schuster's softened star system, for short, as you just told us, aka Dr. Plummer's patented potential. And now we have to call upon Dr. Poisson to provide us with the density that corresponds to this potential.

Alice: But first you have to remind the students of the symmetry assumptions that go into this type of construction.

Bob: Ah yes. We assume spherical symmetry in space, in other words, both potential and density only depend on the distance to the center , independent of the spherical angles :

In addition, we assume that the velocity dispersion is isotropic. In general, spherical symmetry in configuration space (or position space) still allows cylindrical symmetry in velocity space: the distribution function must be a function of energy per unit mass and angular momentum per unit mass . However, by insisting on isotropy we assert spherical symmetry in velocity space as well, which implies:

By the way, the potential is also the potential energy per unit mass -- just like the potential in an electrostatic field, where it is the binding energy per unit charge. To be really specific, take a star with mass at a a position in a star cluster at a distance from the center. In order to let that star escape from the cluster, you have to give it a kinetic energy that is exactly equal to the binding energy of that star with respect to the cluster, namely the absolute value of the potential energy of that star corresponding to its position in the gravitational well of the cluster: .

So this means that the escape velocity for a star at radial distance from the center of the cluster can be derived as follows:

Alice: Good idea to make that very clear, since when you talk about potential and potential energy, it is at first quite easy to get confused between the potential energy of the cluster as a whole and between that of individual particles with respect to the rest of the cluster.

Bob: Well, we don't want to get confused, do we.

Alice: Aha, if you put it that way, I can't resist pointing out another possible confusion. When you wrote down the condition for isotropy, you were right, speaking as a physicist, although a mathematician looking over your shoulder would be greatly confused. Using the same symbol for two expressions that have a different functional form, and what is worse, even have a different number of parameters is definitely a no-no in mathematics.

Bob: But what else could I have written but ?

Alice: A proper mathematical expression would have been , for example. This would imply that as a function is very different from the function . The only connection is that for any choice of a particular value for and a particular value for , the relationship would hold, independently of the value of .

Bob: Hmm. I prefer to say . That makes sense and it feels good. As Janis Joplin would have said, `feeling good is good enough for me.'

Alice: It shows you're a physicist. Just be gentle with the occasional student who may think deeper about these issues than you; she or he may actually have a very good reason to be bothered by the usual glossing over of these questions. After all, it is easy to make mistakes -- for example, mistakes in normalization, if you're not careful about what depends in which way on what.

Bob: I suppose you would have wanted me to put hats on the first two equations as well: and .

Alice: Strictly speaking, yes, but once you have made that point, I prefer to then drop the hats, for simplicity.

Bob: Is that called having your cake and eating it, first confusing me by putting a hat on, and then dropping your hat?

Alice: Maybe that's where the expression `I'll eat my hat' comes from.

1.4. Density

Bob: No comment. Let me go on, and derive the density, given the potential.

To find the density needed to generate a potential, all we have to do is solve Poisson's equation, which in spherical coordinates, under the assumption of spherical symmetry for the potential, takes the relatively simple form

Working out the derivatives, this boils down to:

It is straightforward to work out the first derivative

and the second derivative of the potential

Putting it all together we get:

Alice: It might have been a bit quicker to substitute , given in eq. (
1), directly in eq. (6).

Bob: Probably. But I like to be systematic, and it was nice to get the force on a unit mass, , for free.

So now we can also see what the physical interpretation was of the softened particles that we have been using occasionally: the potential--density pair of a softened point particle is Plummer's pair:

Alice: That looks right. It has the right dimensions: energy for and mass per cubic length, in other words, mass divided by volume for .

Bob: To get more of a feeling for the behavior of these two functions, I find it helpful to factor out the radial distance squared in units of the structural length squared:

or the opposite: the inverse radial distance squared in units of the structural length squared:

1.5. Physical Interpretations

Alice: Yes, that makes it easy to see immediately how the functions behave for small and for large radii. In fact, to bring out the physics even more, I suggest that for your first choice, you introduce the central potential and the central density :


It is then clear right away how the softening works: instead of the infinitely deep potential well of a point mass, the bottom of the well corresponds to the surface potential of an object with mass equal to the total mass of the particle, and radius equal to the structural length. Similarly, the central density is exactly the density that such an object would have. It is nice that it all corresponds to such a simple picture, not only in order of magnitude, but even without any correction factor. That makes it easy to remember the formulas in this way.

Bob: Ah, yes, that is an even nicer way to write things. But I don't like to remember formulas, easy or not! As long as I keep my notes, I can always quickly look them up.

Alice: Or, in practice, rederive them. Don't try to make me believe that you, a), actually keep all your notes, and, b), have a way to find them when you need them!

Bob: Hmmm, well, yes, I'm afraid I often fail at both.

Alice: So do I, and most everyone I know. Actually, this in and of itself is already a good reason for us to write these notes out in book form, as we have started doing. And the more detail, the better: I hate having to spend a few pages of pen and paper work in order to get from one line to the next in a text book. Back in the days that printed paper was expensive, that might have been a good thing, but I prefer to save time, rather than paper. Actually, we're saving paper too, when we just put it all up on the net. So let's keep these notes, in full detail, just the way we're deriving them, in real time.

Bob: Fine with me. And I must admit, since we started writing these notes, I've gone back to them already several times, to look up things that I knew I knew just a few weeks ago, but which had started to slip out of my working memory.

Coming back to your recasting of the potential-density expressions, I suppose we can do something similar for my second choice, but there we are dealing with quantities at infinity, for which the vanishes.

Alice: Indeed, but instead of constants, we are now dealing with functions, basically because we cannot reach infinity, while we can reach the center, at zero distance. What you have factored out are the asymptotic behaviors of the function when you go out to very large radii;

What you have defined, effectively, are the asymptotic form of the potential as the Kepler potential of a point mass, and the asymptotic form of the density as the leading power law function, proportional to the inverse fifth power of the radius:

Bob: I thought I had done something interesting, even though I was mainly doodling, enjoying the equations I had derived by looking at them from different directions. But pretty as they may be, I feel it's time to actually construct a model along Plummer's line. Let me see what I can do, before we next see each other again.
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