  Solar system observations have traditionally allowed for very stringent tests
of Einstein's theory of general relativity. We here revisit the possibility of
using these observations to constrain gravitational parity violation as
encapsulated in dynamical Chern-Simons gravity. Working in the small-coupling
and post-Newtonian approximations, we calculate analytically the scalar field
and the gravitomagnetic sector of the gravitational field in the interior and
the exterior of an isolated, weakly-gravitating object with uniform rotation
and a quadrupolar mass deformation. We find that the asymptotic peeling-off
behavior of the exterior fields is consistent with that found for black holes
and neutron stars, as well as for non-relativistic objects, with overall
coefficients that are different and dependent on the structure of the
weak-field source. We then use these fields to explicitly calculate the
dynamical Chern-Simons correction to the spin precession of gyroscopes in orbit
around Earth, which we find to be in the same direction as the Lense-Thirring
effect of General Relativity. We then compare this correction to the spin
precession prediction of General Relativity to the results of the Gravity Probe
B experiment to place a constraint on dynamical Chern-Simons theory that is
consistent with previous approximate estimates. Although we focus primarily on
a single body, our methods can be straightforwardly extended to binary systems
or N-bodies.
