circle packing on a truncated hemisphere

Question

I have a truncated hemisphere: z= $\sqrt{ R^2 - x^2 -y^2 } $ for $0 \le z \le (R -z_{top})$, where $z_{top} \approx R/3$ or $R/4$. I'd like to cover this truncated hemisphere with circles of radius $r_c$, where $r_c \ll R$. This is a piece of machinery I'm making. I've covered the hemisphere with squares, and am now considering circles. I expect the hemisphere to contain say 100 or more circles. Note that the number of circles is not known yet. I've found lots of information on how to cover spheres with a fixed number of circles, but that number is always low, and there's always the poles, which I don't have. There is also lots of information about covering planar shapes with circles, and stuff with JavaScript, which I have never used. I don't need a theorem or something provably best, I'd just like a guide to positioning the circles for maximum coverage. I'd appreciate any help I can get.