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** 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!

** 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!

** 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:

** 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 :

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

** 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.

** 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:

** 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:

** 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:

** 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
:

** 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;

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