Tuesday, August 28, 2012

Elemental distribution part 1

We're examining a list of points in which we've accumulated celestial material. The next step is to examine each of the points in turn and determine what elements are going to appear there, and in what concentrations. The basic idea is to distribute the elements according to two parameters: the atomic "weight" of an element and the relative mass of the point we're examining.

The simplest example would be a distribution represented by a diagonal line on a graph where the lightest elements have the highest concentration and the heaviest elements have the least. Like this:

As the elements get heavier, their concentration drops. Easy. This is a line given by f(x) = -x + C (C being some useful offset > 0) and I could give my elements a concentration by simply assigning them a value from that formula with x being the element's number (an analogue for its weight).

But we want to add some more complexity. I'd like that line (without the C factor) to rotate around the origin if the structure were especially heavy, meaning that the concentration of elements in a high-density object would tend towards heavier elements (since more stuff = more pressure = heavier elements created), like this:

And that is just f(x) = x.

So I could say that the elemental concentrations in the least dense object in my galaxy would look like graph #1, and the densest object would look like graph #2. What about everything in between (ie, where most of my points will be)? For example, from these two graphs we would then expect the plot of an object in the exact middle of all densities to look like this:

With every element having a distribution equal to every other element.

Well, I could try performing a matrix rotation on my line based on a given point's density relative to the densest or sparsest points in the galaxy, and then take the integral of the resulting line under the area where the current element would lie, but that would leave me with the difficult problem of expressing the formula of the newly rotated line, and that's not a problem I really want to tackle. Instead, I'll simply use the density of the current point to determine the end-points of my line on the x-axis and then interpolate between those points for each element to get my density (y) values.

Okay, that works nicely, but now what I've got is a bunch of straight lines determining my distributions which is totally unrealistic, so the next step is to introduce some randomness into the concentrations. I will go into this next.

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