I was trying to determine the probability of star collision during a galaxy collision, and came up with this problem :
A sphere of radius $R$ and origin $O$ (the Galaxy) has an isotropic Gaussian distribution of dots (center of stars) per units of volume inside it : $n(r)=n_0e^{-\frac{r^2}{2\sigma^2}}$
such as its integral over the whole volume is $N$ (number of dots contained in the sphere)
Is there a closed form for $n_{cs}(r)$, which is the distribution of dots projected orthogonally to the cross section at the origin of the sphere (disk of radius $\pi R^2$) ? Is it still a Gaussian ?
I have a hard time computing it since when I try to resolve this problem, I find myself having to integrate the distribution $n(r)$ over a cylinder which is orthogonal to the plane of the cross section at the origin, and can't find the bounds in terms of $r,\theta,\phi$ or $x,y,z$. Maybe there is an easier way with another system of coordinates I do not know ?
Truncating the Gaussian to a sphere makes things ugly and complicated, but if you can neglect the truncation and pretend that you have an isotropic Gaussian distribution throughout all of space, the answer is yes: The distribution along each axis individually, and in all coordinate planes, is again Gaussian. This is due to the factorization property
$$ \mathrm e^{-\frac{r^2}{2\sigma^2}}=\mathrm e^{-\frac{x^2}{2\sigma^2}}\mathrm e^{-\frac{y^2}{2\sigma^2}}\mathrm e^{-\frac{z^2}{2\sigma^2}}\;, $$
which shows that the individual coordinates are independent.