  We investigate the limits of ground-based astrometry with adaptive optics
using the core of the Galactic globular cluster M5. Adaptive optics systems
provide near diffraction-limit imaging with the world's largest telescopes. The
substantial improvement in both resolution and signal-to-noise ratio enables
high-precision astrometry from the ground. We describe the dominant systematic
errors that typically limit ground-based differential astrometry, and enumerate
observational considerations for mitigating their effects. After implementing
these measures, we find that the dominant limitation on astrometric performance
in this experiment is caused by tilt anisoplanatism. We then present an optimal
estimation technique for measuring the position of one star relative to a grid
of reference stars in the face of this correlated random noise source. Our
methodology has the advantage of reducing the astrometric errors as the square
root of time and faster than the square root of the number of reference stars
-- effectively eliminating noise caused by atmospheric tilt to the point that
astrometric performance is limited by centering accuracy. Using 50 reference
stars we demonstrate single-epoch astrometric precision of ~ 1 mas in 1 second,
decreasing to < 100 microarcseconds in 2 minutes of integration time at the
Hale 200-inch telescope. We also show that our astrometry is accurate to <~ 100
microarcseconds for observations separated by 2 months. Finally, we discuss the
limits and potential of differential astrometry with current and next
generation large aperture telescopes. At this level of accuracy, numerous
astrometric applications become accessible, including planet detection,
astrometric microlensing signatures, and kinematics of distant Galactic stellar
populations.
