  Spherical harmonic expansions (SHEs) play an important role in most of the
physical sciences, especially in physical geodesy. Despite many decades of
investigation, the large order behavior of the SHE coefficients, and the
precise domain of convergence for these expansions, have remained open
questions. These questions are settled in the present paper for generic
planets, whose shape (topography) may include many local peaks, but just one
globally highest peak. We show that regardless of the smoothness of the density
and topography, short of outright analyticity, the spherical harmonic expansion
of the gravitational potential converges exactly in the closure of the exterior
of the Brillouin sphere (The smallest sphere around the center of mass of the
planet containing the planet in its interior), and convergence below the
Brillouin sphere occurs with probability zero. More precisely, such
over-convergence occurs on zero measure sets in the space of parameters. A
related result is that, in a natural Banach space, SHE convergence of the
potential below the Brillouin sphere occurs for potential functions in a
subspace of infinite codimension (while any positive codimension already
implies occurrence of probability zero). Provided a certain limit in Fourier
space exists, we find the leading order asymptotic behavior of the coefficients
of SHEs.
  We go further by finding a necessary and sufficient condition for convergence
below the Brillouin sphere, which requires a form of analyticity at the highest
peak, which would not hold for a realistic celestial body. Namely, a
longitudinal average of the harmonic measure on the Brillouin sphere would have
to be real-analytic at the point of contact with the boundary of the planet. It
turns out that only a small neighborhood of the peak is involved in this
condition.
