  The densest local packings of N identical nonoverlapping spheres within a
radius Rmin(N) of a fixed central sphere of the same size are obtained using a
nonlinear programming method operating in conjunction with a stochastic search
of configuration space. Knowledge of Rmin(N) in d-dimensional Euclidean space
allows for the construction both of a realizability condition for pair
correlation functions of sphere packings and an upper bound on the maximal
density of infinite sphere packings. In this paper, we focus on the
two-dimensional circular disk problem. We find and present the putative densest
packings and corresponding Rmin(N) for selected values of N up to N = 348 and
use this knowledge to construct such a realizability condition and upper bound.
We additionally analyze the properties and characteristics of the maximally
dense packings, finding significant variability in their symmetries and contact
networks, and that the vast majority differ substantially from the triangular
lattice even for large N. Our work has implications for packaging problems,
nucleation theory, and surface physics.
