# I Plummer distribution of stars

1. Mar 15, 2016

### jimbo007

Hi all,
I refer to the following pdf document, in particular the appendix:

End goal is to distribute N stars each having mass m (looks like equal mass stars is the easiest scenario) within a box who has each side of length $3 \times 10^{17} \text{m}$ so that the total mass is M. I think I understand that you can't get all stars in the box but as long as most are in the box that should be close enough.

There are some parts of this document I don't understand but will take on faith that it is true.

I do understand how the position (x,y,z) and velocity (u,v,w) are calculated but not sure how to convert them to metres and metres/sec.

Does anyone know how to convert each stars mass m, position (x,y,z) and velocity (u,v,w) into kg, m and m/s?

Thanks!

2. Mar 16, 2016

### jimbo007

I probably should at least put a few more equations in to make things a bit easier. Plummer's density is given by:
$$\rho(r) = \frac{3M}{4\pi R^3}\left( 1+ \left(\frac{r}{R} \right)^2 \right)^{-\frac{5}{2}}$$
where
M = total mass of star cluster
R = magic scaling parameter

Now let $X_{n}$ be a uniformly distributed random number between 0 and 1. (x, y, z) is the position of the star and (u, v, w) is the velocity of the star

$$\mathbf{x} = \begin{pmatrix}x \\ y \\ z\end{pmatrix} \\ \dot{\mathbf{x}} = \begin{pmatrix} u \\ v \\ w \end{pmatrix}$$
then
$$r = \left(X_{1}^{-\frac{2}{3}} - 1 \right)^{-\frac{1}{2}} \\ z = \left( 1-2X_{2} \right) r$$
x, y, u, v and w and calculated in a similar fashion. The last paragraph in the pdf says to multiply these variables by a few numbers but it's not clear to me how to convert z to metres