It is well known that, given a sphere, the maximum number of identical spheres that we can pack around it is exactly 12, corresponding to a face centered cubic or hexagonal close packed lattice.
My question is: given a sphere of radius $R$, how many spheres of radius $r<R$ can we closely pack around it?
With disks, the problem is rather easy to solve. Indeed, with reference to the picture at the bottom, we can see that we must have
$$\theta = \frac{2 \pi} n = 2 \arctan \left( \frac r {\sqrt{R^2+2 R r}} \right)$$
from which
$$n = \left \lfloor \frac \pi {\arctan \left( \frac r {\sqrt{R^2+2 R r}} \right)}\right \rfloor$$
The last expression gives the correct result for $R=r$, namely $n=6$ (hexagonal lattice). Moreover, when $R \gg r$, we get
$$n \simeq \left \lfloor \frac {\pi R} {r}\right \rfloor$$
which is completely reasonable.
How can I tackle the same problem in the 3D case (spheres)?
It is clear that for $R \gg r$ we must get
$$n \simeq \left \lfloor \frac {4 \pi R^2} {\pi r^2}\right \rfloor$$
and also that we must have $n(R=r)=12$.
Any hint/suggestion is appreciated.
Two small spheres touching the large sphere fail to intersect if and only if their projections onto the surface of the large sphere do not overlap. Since there is a bijective map between small sphere radii and the radius of the corresponding circular discs on the sphere they get projected to, this problem is equivalent to finding the maximum number of points on a sphere that are mutually at a distance at least $d>0$ from each other.
Equivalently, for a fixed number of points $k$, we can ask for the maximum $d$ such that $k$ points can be so arranged; under this framing, the problem is known as the Tammes problem.
In general packing problems like this are incredibly difficult and have rather ugly solutions; you will find no elegant formulas, regular constructions, or easy proofs for the optimal solutions here, outside of isolated unusually nice cases like the case where the points form an inscribed Platonic solid with triangular faces.
That said, there exist pretty good numerical solutions to these problems for small values of $k$, and of course in the limit things will approach the packing density of circles in the plane at $\frac{\pi\sqrt{3}}{6}\approx 0.906$.
You can find a writeup on the problem here, but many others are available by searching online.