  In order to extract physical parameters from the waveform of an
extreme-mass-ratio binary, one requires a second-order--accurate description of
the motion of the smaller of the two objects in the binary. Using a method of
matched asymptotic expansions, I derive the second-order equation of motion of
a small, nearly spherical and non-rotating compact object in an arbitrary
vacuum spacetime. I find that the motion is geodesic in a certain locally
defined effective metric satisfying the vacuum Einstein equation through second
order, and I outline a method of numerically calculating this effective metric.
