  We study the constraining power of a high-precision measurement of the
gravity field for Uranus and Neptune, as could be delivered by a low periapse
orbiter. Our study is practical, assessing the possible deliverables and
limitations of such a mission with respect to the structure of the planets. Our
study is also academic, assessing in a general way the relative importance of
the low order gravity, high order gravity, rotation rate, and moment of inertia
(MOI) in constraining planetary structure. We attempt to explore all possible
interior density structures of a planet that are consistent with hypothetical
gravity data, via MCMC sampling of parameterized density profiles. When the
gravity field is poorly known, as it is today, uncertainties in the rotation
rate on the order of 10 minutes are unimportant, as they are interchangeable
with uncertainties in the gravity coefficients. By the same token, when the
gravity field is precisely determined the rotation rate must be known to
comparable precision. When gravity and rotation are well known the MOI becomes
well-constrained, limiting the usefulness of independent MOI determinations
unless they are extraordinarily precise. For Uranus and Neptune, density
profiles can be well-constrained. However, the non-uniqueness of the relative
roles of H/He, watery volatiles, and rock in the deep interior will still
persist with high-precision gravity data. Nevertheless, the locations and
magnitudes (in pressure-space) of any large-scale composition gradient regions
can likely be identified, offering a crucially better picture of the interiors
of Uranus or Neptune.
