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, and I appreciate everyone's time.

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.