  Given a sphere with Bartnik data close to that of a round sphere in Euclidean
3-space, we compute its Bartnik-Bray outer mass to first order in the data's
deviation from the standard sphere. The Hawking mass gives a well-known lower
bound, and an upper bound is obtained by estimating the mass of a static vacuum
extension. As an application we confirm that in a time-symmetric slice
concentric geodesic balls shrinking to a point have mass-to-volume ratio
converging to the energy density at their center, in accord with physical
expectation and the behavior of other quasilocal masses. For balls shrinking to
a flat point we can also compute the outer mass to fifth order in the
radius---the term is proportional to the Laplacian of the scalar curvature at
the center---but our estimate is not refined enough to identify this term at a
point which is merely scalar flat. In particular it cannot discern
gravitational contributions to the mass.
