3. Sprinkling Particles in Space
Alice: Now I understand what your code is doing, except for a few crucial
lines. First there is the one-liner about choosing the distance
between a new star and the center of the star cluster:
Can you tell me what this expression means, and how it is derived?
It must somehow be related to the density distribution
Bob: They describe their technique in a few words, and I had to read
those words carefully, and do some head scratching, to figure out what
it meant. But as always, once you see it, it is really easy. Let me
try to summarize it in my own words.
First we introduce the cumulative mass distribution
In other words, it will see a fraction
The ranking of each particle, in terms of the enclosed mass, is random
and uniform in the mass fraction. In other words,
So here is the idea: spin a random number generator in order to obtain
a a random number
Alice: That sounds straightforward. Can you show me the expressions
you found for
Bob: I simply took their expressions. They use a system of units in which
the total mass M, the gravitational constant G and the structural length
scale a that we used above are all unity. The mass enclosed within
a radius r then becomes:
Alice: Hmm. You didn't check whether they had done their math
correctly?
Bob: No need to. This is a paper by Aarseth, Henon, and Wielen.
Besides, it is thirty years old and has been cited zillions of times
by others. I'm sure this is a result that can be trusted.
Alice: I don't like to accept things on trust, no matter what the
authority may be behind it. Not that I expect them to be wrong,
I agree that that would be highly improbable. Still, I would feel
much better if we derive the results ourselves. Besides, if we work
it out now, we can both use those notes when we have to teach it in
class to the students. Better still, we just put it into the material
we prepare for them on the web.
Bob: Okay, if you like. Your turn, though, I already derived the
density. Do you want to use a package from symbolic integration?
Differentiation is easy enough by hand, but I must admit, I'm a bit
rusty in my integration.
Alice: So am I, and that is a really good reason to do it with pen and
paper, tempting as it is to use a symbolic package. Okay. I'll start
with the density you derived:
which gives us:
This gives us:
Bob: Well, if you are rusty in your integrations, then I don't know what
to call myself. Nice job! It is always surprising to me how the result
of that type of calculation can come out in such a simple form.
Alice: There probably is a good physical reason for it to be this simple.
Let's think. I started with density, something that you had found by
differentiation, and then I integrated the product of the density and
the geometric opening angle factor of
Bob: Hey, wait a minute. I found the density by integrating alright, but
in the following way, using Poisson's equation:
Alice: It does . . . Hey, I could have started there! I could have written:
Alice: Not only that, here is the physical meaning we were looking for!
You also mentioned that the gradient of the potential is the gravitational
force, apart from a minus sign. So what this equation is telling us, is
simply that the physical force is proportional to mass and inversely
proportional to the radius squared: Newton's gravity! We could have
started that way. The magnitude of the force on a particle with mass
and also equal to:
since
Bob: You're right. If we would have started with those two lines, we
could have written
right away. And with the expression I wrote down yesterday,
this would have given us:
Alice: Quite a bit faster than my juggling of integrals! We could have
used a healthy dose of physical intuition, before embarking on that lengthy
computation.
Bob: Now that we have found the radius dependence of the cumulative mass,
we only have to invert that relationship, to get the dependence of cumulative
mass on radius. That shouldn't be too hard. However, I'm getting tired of
carrying along the factors
Alice: But you'd better explain your students how you can restore the
correct factors of
Bob: Good point. It takes a while to learn to think in terms of
dimensionless quantities, and to transform easily and confidently
between those and the corresponding physical quantities. In this
case, I can just point out that the dimensionless quantity
Bob: Yes, students do indeed ask such questions! It just means that they
have to practice more with simple examples, until it comes to them naturally.
Alice: Well, that is not really answering the question. My answer would be
to introduce two different sets of symbols, to remove the confusion between the
physical quantities and the dimensionless quantities.
Bob: I won't stop you!
3.1. Choosing a Distance
radius = 1.0 / sqrt( rand ** (-2.0/3.0) - 1.0)
, which you have already derived from the potential.
How exactly do Aarseth, Henon and Wielen use the density for particle
sprinkling?
from the center. When we create a
new star, and place it at radius 
, that star will see
an amount of mass 
of the cluster at positions closer
to the center than its own position, and therefore an amount of mass
at positions further from the center.
of the
total mass inside its radial position. Now that fraction could be
anything between 0 and 1. It will be 0 if the particle is placed
exactly in the center, and it will approach 1 if the particle is
placed very far away, reaching 1 when the particle is placed at
infinity.
will be a random value between 0 and 1, with each value equally likely.
, with 
,
and we consider that to be the fractional mass contained within the
radius 
of particle 
. So all we know is
that 
, but what we need to know is
itself. So the procedure is to invert (18)
to obtain a function 
, and then life is simple:
.
3.2. Following the Recipe
and 
?
radius = 1.0 / sqrt( rand ** (-2.0/3.0) - 1.0)
3.3. Cumulative Mass
appears in the integrand only in terms of
the combination 
, so a natural change of variables is:
in the integrand. I'll use integration by parts to
lower the power:
3.4. Physical Intuition
. Apart from
that factor, integration and differentiation would have canceled. Pity.
you were looking for?
at distance 
from the center is of course:
everywhere.
3.5. An Intuitive Derivation
and 
which we
may as well consider to be the physical units used for measuring
and 
, so that the latter are used as
dimensionless parameters. Setting 
, we can write:
and 
, otherwise they
will think that you were cutting corners.
has to be replaced by 
, and the
dimensionless quantity 
has to be replaced by
. We then get:
by the length scale 
?
and 
are physical
quantities, while 
is the dimensionless quantity
connecting them. Similarly, 
and 
are
physical quantities, while 
is the dimensionless quantity
giving the variable quantity 
in terms of the mass unit
. We can then write:
