Question
How many spherical caps of height $h$ and base circle radius $a$ can cover a sphere $\mathbb S $ of radius $R \quad (R \gg a)$?
What I have thought so far
Since the area of the sphere is $S_{\mathbb S} = 4 \pi R^2$ and the surface area of the spherical cap (all the faces) is $S_c = \pi (a^2 + h^2)$ then the number $N$ of spherical caps needed to fully cover $\mathbb S$ is greater than $$N_0 = \bigg \lfloor \frac {4 R^2}{h^2} \bigg \rfloor$$ because the shapes of the spherical caps cannot be deformed in order to cover uncovered areas on the sphere.
Define a density function $$\varrho (N) = \frac{A_{\mathrm {overlap}} (N)}{4 \pi r^2}$$ and try to minimize it $\forall \varrho \geq 0 $. Note that $A_{\mathrm {overlap}}(N)$ denotes the total overlapping area for some given number of caps, $N > N_0$
PS Finding the overlapping area function will be a great help.