TITLE: Optimal packing of spheres tangent to a central sphere
QUESTION [5 upvotes]: Please consider a central, ordinary 2-sphere $S_1$, of some radius $r_1$, and a second ordinary sphere, $S_2$, of radius $r_2$, where $r_2 \leq r_1$.     
My question concerns optimal values for the number of spheres of type $S_2$ that can be packed in three-dimensional space so that they are non-overlapping and tangent to $S_1$.  Is there an analytical result for optimal packing a function of the ${r_2 \over r_1}$, or are there subsets of cases that are solved with methods beyond something like simulating annealing?  Does it simplify the problem to apply the further restriction that $r_2 << r_1$?  
I've been having trouble finding answers with a literature search, particularly for the latter situation where $r_2 << r_1$, and I appreciate everyone's time.

REPLY [3 votes]: Assuming w.l.o.g. that $r_1=1$, one can guess the asymptotic behaviour of the kissing number as $r_2\to 0$ as a function $f(r_2)$.  Consider a radial projection of the smaller balls onto the surface of the large (closed) ball (where the projection is w.r.t. the centre of the large ball).  As $r_2\to 0$, this projection will locally approximate a packing of disks in the plane, and the density of the projection of an optimal packing should approximate the density of an optimal packing of unit disks in the plane.  This density is well known to be $\frac{\pi}{2\sqrt{3}}$.  Since the projection of each small ball covers a surface area of approximately $\pi r_{\kern-1pt2}^2$ of the unit ball, and the unit ball has surface area $4\pi$, we should have for very small $r_2$
$$
f(r_2)\pi r_{\kern-1pt2}^2 \approx \frac{\pi}{2\sqrt{3}}4\pi~,
$$
or
$$
\lim_{r_2\to 0} ~f(r_2) =  \frac{2\pi}{\sqrt{3}\cdot r_{\kern-1pt2}^{2}}~.
$$