  A Sagnac atom interferometer can be constructed using a Bose-Einstein
condensate trapped in a cylindrically symmetric harmonic potential. Using the
Bragg interaction with a set of laser beams, the atoms can be launched into
circular orbits, with two counterpropagating interferometers allowing many
sources of common-mode noise to be excluded. In a perfectly symmetric and
harmonic potential, the interferometer output would depend only on the rotation
rate of the apparatus. However, deviations from the ideal case can lead to
spurious phase shifts. These phase shifts have been theoretically analyzed for
anharmonic perturbations up to quartic in the confining potential, as well as
angular deviations of the laser beams, timing deviations of the laser pulses,
and motional excitations of the initial condensate. Analytical and numerical
results show the leading effects of the perturbations to be second order. The
scaling of the phase shifts with the number of orbits and the trap axial
frequency ratio are determined. The results indicate that sensitive parameters
should be controlled at the $10^{-5}$ level to accommodate a rotation sensing
accuracy of $10^{-9}$ rad/s. The leading-order perturbations are suppressed in
the case of perfect cylindrical symmetry, even in the presence of anharmonicity
and other errors. An experimental measurement of one of the perturbation terms
is presented.
