# Normal distribution and star density in a Galaxy

1. May 19, 2013

### corndog16

Hey all.

I'm working on a personal programming project where I'm attempting to simulate (to a small degree) a galaxy. And I have come across a decent 2D density map for a spiral galaxy. This map (array actually) defines a 128x128 grid of values between 0 and 255 representing the frequency of stars found in that 'sector' of space.

I want to expand this density map into the 3rd dimension and I figure a good way to do that would be to define the height of the galaxy as being n sectors tall and then use a normal distribution curve to distribute the density value (something between 0 and 255) vertically across those 5 sectors.

My question is, how would I take a number, say 234, and divide it among n 'boxes' or 'sectors' so that it has a normal curve type distribution?
I attempted to Google this, but I found that what was coming up did not describe what I was looking for so I'm thinking that I don't currently have the vocabulary to describe this particular intent.

2. May 20, 2013

### Stephen Tashi

A normal distribution has two parameters, the mean $\mu$ and the standard deviation $\sigma$. Lets say we have 128 boxes. Imagine the boxes as spaces between a ruler that has marks at 0,1,2,..128. Assuming you want the middle of the distribution to be at the middle of the ruler, set $\mu = 64$.

Then you have to decide on a value for $\sigma$. A normal distribution has non-zero values from $- \infty$ to $+\infty$, so you have to decide what part of the distribution you are going to leave out or you have to settle for a distribution that doesn't exactly match a normal distribution by forcing all of the stars to be within the 128 boxes.

For most programming languages, you can find a library with a function that give the values of the cumulative normal distibution. Let's assume that such a function is

$F(\mu,\sigma,x) = \int_{-\infty}^x f(\mu,\sigma,x) dx$ where $f(\mu,\sigma,x)$ is the density of a normal distribution.

The fraction of stars between marks $A$ and $B$ on the ruler is

$F(64,\sigma,B) - F(64,\sigma,A)$.

So the fraction of stars between the 0 and 128 marks is

$M = F(64,\sigma,128) - F(64,\sigma,0)$

The fraction that are outside those marks is $1 - M$.

The fraction of stars in the kth box is $s_k = F(64,\sigma,k) - F(64,\sigma,k-1)$. If you want to "renormalize" so that all the stars fall in the marks between 0 and 128, you can make the fraction of stars in the kth box $\frac{s_k}{1-M}$

For 234 total stars, the number of stars in the kth box will be $(s_k)(234)$. So it won't always be a whole number. To convert to a while number, you'll have to decide how to round things off. (If you want the grand total to remain 234, you'll have to round off in a special way!)

The above calculations don't determine a unique value for $\sigma$. You'll have to try various values to find one that you like if your goal is only to have a pleasing picture. If you are doing a physical simulation of moving stars, we'll have to investigate the problem further.

3. Nov 27, 2013

### davidhayes

Hi,
Any chance you could post your source for the 2D Density Map you have? I'm looking for exactly that
Thanks
Dave