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.



Optimal packing of spheres tangent to a central sphere
I believe what you are seeking is sometimes known as a 
*spherical packing* or a *spherical code*. Here is 
[the MathWorld article](http://mathworld.wolfram.com/SphericalCode.html) on the topic.
Here is [Neil Sloane's webpage](http://neilsloane.com/packings/index.html) on the topic, including "putative
optimal packings" up to $n=130$ spheres.
See also [integer sequence A126195](https://oeis.org/A126196) for the
"Conjectured values for maximal number of solid spheres of radius 1 that can be rolled all in touch with and on the outside surface of a sphere of radius $n$."


There are bounds, but no "analytical result" of the type for which you might be hoping.


**Tangential Addendum**.
There is an interesting variant of the sphere-packing kissing number
that seems little known, and for which the bounds are wide enough to invite further
work. It is called *Hornich's Problem*: What is the fewest number of non-overlapping
closed unit balls that can radially hide a unit ball $B$, in the sense that every ray from the
center of $B$ intersects at least one of the surrounding balls?
In $\mathbb{R}^3$, it is known that at least 30 balls are needed, and 42 suffice.
So rather different than the kissing number 12!


Described [on p.117](https://books.google.com/books?id=WehCspo0Qa0C&pg=PA117) of
*Research Problems in Discrete Geometry*,
Brass, Peter, Moser, William O. J., Pach, János,
2005.
([Springer book link](https://www.springer.com/us/book/9780387238159).)


