I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching a central unit ball.

That is, center the unit ball $B \subset \mathbb{R}^3$ at the origin and define $k_{s}(\eta)$ to be the maximum number of balls of radius $\eta$ which can kiss $B$ and form a solid packing (non-overlapping interiors). How can you upper bound $k_{s}(\eta)$ for $\eta >0$? A remarkable case of note is $k_{s}(1) = 12$ which corresponds to the solution of the Newton-Gregory problem.